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Thomas Calculus

George B. Thomas, Jr.

Chapter 7

Transcendental Functions

Educators


Problem 1

Which of the functions graphed are one-to-one, and which are not?

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Problem 2

Which of the functions graphed are one-to-one, and which are not?

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Problem 3

Which of the functions graphed are one-to-one, and which are not?

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Problem 4

Which of the functions graphed are one-to-one, and which are not?

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Problem 5

Which of the functions graphed are one-to-one, and which are not?

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Problem 6

Which of the functions graphed are one-to-one, and which are not?

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Problem 7

Determine from its graph if the function is one-to-one.
$$f(x)=\left\{\begin{array}{ll}{3-x,} & {x<0} \\ {3,} & {x \geq 0}\end{array}\right.$$

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Problem 8

Determine from its graph if the function is one-to-one.
$$f(x)=\left\{\begin{array}{ll}{2 x+6,} & {x \leq-3} \\ {x+4,} & {x>-3}\end{array}\right.$$

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Problem 9

Determine from its graph if the function is one-to-one.
$$f(x)=\left\{\begin{array}{ll}{1-\frac{x}{2},} & {x \leq 0} \\ {\frac{x}{x+2},} & {x>0}\end{array}\right.$$

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Problem 10

Determine from its graph if the function is one-to-one.
$$f(x)=\left\{\begin{array}{ll}{2-x^{2},} & {x \leq 1} \\ {x^{2},} & {x>1}\end{array}\right.$$

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Problem 11

Shows the graph of a function $y=f(x)$. Copy the graph and draw in the line $y=x .$ Then use symmetry with respect to the line $y=x$ to add the graph of $f^{-1}$ to your sketch. (It is not necessary to find a formula for $f^{-1}.$ ) Identify the domain and range of $f^{-1}$.

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Problem 12

Shows the graph of a function $y=f(x)$. Copy the graph and draw in the line $y=x .$ Then use symmetry with respect to the line $y=x$ to add the graph of $f^{-1}$ to your sketch. (It is not necessary to find a formula for $f^{-1}.$ ) Identify the domain and range of $f^{-1}$.

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Problem 13

Shows the graph of a function $y=f(x)$. Copy the graph and draw in the line $y=x .$ Then use symmetry with respect to the line $y=x$ to add the graph of $f^{-1}$ to your sketch. (It is not necessary to find a formula for $f^{-1}.$ ) Identify the domain and range of $f^{-1}$.

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Problem 14

Shows the graph of a function $y=f(x)$. Copy the graph and draw in the line $y=x .$ Then use symmetry with respect to the line $y=x$ to add the graph of $f^{-1}$ to your sketch. (It is not necessary to find a formula for $f^{-1}.$ ) Identify the domain and range of $f^{-1}$.

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Problem 15

Shows the graph of a function $y=f(x)$. Copy the graph and draw in the line $y=x .$ Then use symmetry with respect to the line $y=x$ to add the graph of $f^{-1}$ to your sketch. (It is not necessary to find a formula for $f^{-1}.$ ) Identify the domain and range of $f^{-1}$.

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Problem 16

Shows the graph of a function $y=f(x)$. Copy the graph and draw in the line $y=x .$ Then use symmetry with respect to the line $y=x$ to add the graph of $f^{-1}$ to your sketch. (It is not necessary to find a formula for $f^{-1}.$ ) Identify the domain and range of $f^{-1}$.

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Problem 17

\begin{equation}
\begin{array}{l}{\text { a. Graph the function } f(x)=\sqrt{1-x^{2}}, 0 \leq x \leq 1 . \text { What symmetry }} \\ \quad {\text { does the graph have? }} \\ {\text { b. Show that } f \text { is its own inverse. (Remember that } \sqrt{x^{2}}=x \text { if }} \\ \quad {\quad x \geq 0 . )}\end{array}
\end{equation}

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Problem 18

\begin{equation}
\begin{array}{l}{\text { a. Graph the function } f(x)=1 / x . \text { What symmetry does the }} \\ \quad {\text { graph have? }} \\ {\text { b. Show that } f \text { is its own inverse. }}\end{array}
\end{equation}

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Problem 19

Gives a formula for a function $y=f(x)$ and shows the graphs of $f$ and $f^{-1} .$ Find a formula for $f^{-1}$ in each case.
$$f(x)=x^{2}+1, \quad x \geq 0$$

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Problem 20

Gives a formula for a function $y=f(x)$ and shows the graphs of $f$ and $f^{-1} .$ Find a formula for $f^{-1}$ in each case.
$$f(x)=x^{2}, \quad x \leq 0$$

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Problem 21

Gives a formula for a function $y=f(x)$ and shows the graphs of $f$ and $f^{-1} .$ Find a formula for $f^{-1}$ in each case.
$$f(x)=x^{3}-1$$

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Problem 22

Gives a formula for a function $y=f(x)$ and shows the graphs of $f$ and $f^{-1} .$ Find a formula for $f^{-1}$ in each case.
$$f(x)=x^{2}-2 x+1, \quad x \geq 1$$

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Problem 23

Gives a formula for a function $y=f(x)$ and shows the graphs of $f$ and $f^{-1} .$ Find a formula for $f^{-1}$ in each case.
$$f(x)=(x+1)^{2}, \quad x \geq-1$$

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Problem 24

Gives a formula for a function $y=f(x)$ and shows the graphs of $f$ and $f^{-1} .$ Find a formula for $f^{-1}$ in each case.
$$f(x)=x^{2 / 3}, \quad x \geq 0$$

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Problem 25

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=x^{5}$$

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Problem 26

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=x^{4}, \quad x \geq 0$$

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Problem 27

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=x^{3}+1$$

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Problem 28

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=(1 / 2) x-7 / 2$$

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Problem 29

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=1 / x^{2}, \quad x>0$$

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Problem 30

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=1 / x^{3}, \quad x \neq 0$$

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Problem 31

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=\frac{x+3}{x-2}$$

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Problem 32

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=\frac{\sqrt{x}}{\sqrt{x}-3}$$

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Problem 33

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=x^{2}-2 x, \quad x \leq 1(\text { Hint: Complete the square.) }$$

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Problem 34

Gives a formula for a function $y=f(x) .$ In each case, find $f^{-1}(x)$ and identify the domain and range of $f^{-1}$. As a check, show that $f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x.$
$$f(x)=\left(2 x^{3}+1\right)^{1 / 5}$$

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Problem 35

\begin{equation}
\begin{array}{l}{\text { a. Find } f^{-1}(x)} \\ {\text { b. Graph } f \text { and } f^{-1} \text { together. }} \\ {\text { c. Evaluate } d f / d x \text { at } x=a \text { and } d f^{-1} / d x \text { at } x=f(a) \text { to show that }} \\ \quad {\text { at these points } d f^{-1} / d x=1 /(d f / d x) .}\end{array}
\end{equation}
\begin{equation}f(x)=2 x+3, \quad a=-1\end{equation}

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Problem 36

\begin{equation}\begin{array}{l}{\text { a. Find } f^{-1}(x)} \\ {\text { b. Graph } f \text { and } f^{-1} \text { together. }} \\ {\text { c. Evaluate } d f / d x \text { at } x=a \text { and } d f^{-1} / d x \text { at } x=f(a) \text { to show that }} \\ \quad {\text { at these points } d f^{-1} / d x=1 /(d f / d x) .}\end{array}
\end{equation}
\begin{equation}f(x)=(1 / 5) x+7, \quad a=-1\end{equation}

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Problem 37

\begin{equation}\begin{array}{l}{\text { a. Find } f^{-1}(x)} \\ {\text { b. Graph } f \text { and } f^{-1} \text { together. }} \\ {\text { c. Evaluate } d f / d x \text { at } x=a \text { and } d f^{-1} / d x \text { at } x=f(a) \text { to show that }} \\ \quad {\text { at these points } d f^{-1} / d x=1 /(d f / d x) .}\end{array}
\end{equation}
\begin{equation}f(x)=5-4 x, \quad a=1 / 2\end{equation}

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Problem 38

\begin{equation}\begin{array}{l}{\text { a. Find } f^{-1}(x)} \\ {\text { b. Graph } f \text { and } f^{-1} \text { together. }} \\ {\text { c. Evaluate } d f / d x \text { at } x=a \text { and } d f^{-1} / d x \text { at } x=f(a) \text { to show that }} \\ \quad {\text { at these points } d f^{-1} / d x=1 /(d f / d x) .}\end{array}
\end{equation}
\begin{equation}f(x)=2 x^{2}, \quad x \geq 0, \quad a=5\end{equation}

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Problem 39

\begin{equation}\begin{array}{l}{\text { a. Show that } f(x)=x^{3} \text { and } g(x)=\sqrt[3]{x} \text { are inverses of one }} \\ \quad {\text { another. }} \\ {\text { b. Graph } f \text { and } g \text { over an } x \text { -interval large enough to show the }} \\ \quad {\text { graphs intersecting at }(1,1) \text { and }(-1,-1) . \text { Be sure the picture }} \\ \quad {\text { shows the required symmetry about the line } y=x \text { . }} \\ {\text { c. Find the slopes of the tangents to the graphs of } f \text { and } g \text { at }} \\ \quad {(1,1) \text { and }(-1,-1) \text { (four tangents in all). }} \\ {\text { d. What lines are tangent to the curves at the origin? }} \end{array}\end{equation}

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Problem 40

\begin{equation}\begin{array}{l}{\text { a. Show that } h(x)=x^{3} / 4 \text { and } k(x)=(4 x)^{1 / 3} \text { are inverses of one }} \\ \quad {\text { another. }} \\ {\text { b. Graph } h \text { and } k \text { over an } x \text { -interval large enough to show the }} \\ \quad {\text { graphs intersecting at }(2,2) \text { and }(-2,-2) . \text { Be sure the picture }} \\ \quad{\text { shows the required symmetry about the line } y=x \text { . }}\\ {\text { c. Find the slopes of the tangents to the graphs of } h \text { and } k \text { at }} \\ \quad {(2,2) \text { and }(-2,-2) .} \\ {\text { d. What lines are tangent to the curves at the origin? }}\end{array}
\end{equation}

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Problem 41

\begin{equation}
\begin{array}{l}{\text { Let } f(x)=x^{3}-3 x^{2}-1, x \geq 2 . \text { Find the value of } d f^{-1} / d x \text { at }} \\ {\text { the point } x=-1=f(3)}.\end{array}
\end{equation}

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Problem 42

\begin{equation}
\begin{array}{l}{\text { Let } f(x)=x^{2}-4 x-5, x>2 . \text { Find the value of } d f^{-1} / d x \text { at }} \\ {\text { the point } x=0=f(5) .}\end{array}
\end{equation}

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Problem 43

Suppose that the differentiable function $y=f(x)$ has an inverse and that the graph of $f$ passes through the point $(2,4)$ and has a slope of 1$/ 3$ there. Find the value of $d f^{-1} / d x$ at $x=4$.

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Problem 44

Suppose that the differentiable function $y=g(x)$ has an inverse and that the graph of $g$ passes through the origin with slope $2 .$ Find the slope of the graph of $g^{-1}$ at the origin.

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Problem 45

\begin{equation}
\begin{array}{l}{\text { a. Find the inverse of the function } f(x)=m x, \text { where } m \text { is a constant }} \\ \quad {\text { different from zero. }} \\ {\text { b. What can you conclude about the inverse of a function }} \\ \quad {y=f(x) \text { whose graph is a line through the origin with a non-zero}} \\ \quad {\text { slope } m ?}\end{array}
\end{equation}

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Problem 46

Show that the graph of the inverse of $f(x)=m x+b,$ where $m$ and $b$ are constants and $m \neq 0,$ is a line with slope 1$/ m$ and $y$ -intercept $-b / m .$

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Problem 47

\begin{equation}
\begin{array}{l}{\text { a. Find the inverse of } f(x)=x+1 . \text { Graph } f \text { and its inverse }} \\ \quad {\text { together. Add the line } y=x \text { to your sketch, drawing it with }} \\ \quad {\text { dashes or dots for contrast. }} \\ {\text { b. Find the inverse of } f(x)=x+b(b \text { constant). How is the }} \\ \quad {\text { graph of } f^{-1} \text { related to the graph of } f ?} \\ {\text { c. What can you conclude about the inverses of functions whose }} \\ \quad {\text { graphs are lines parallel to the line } y=x ?}\end{array}
\end{equation}

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Problem 48

\begin{equation}\begin{array}{l}{\text { a. Find the inverse of } f(x)=-x+1 . \text { Graph the line }} \\ \quad {y=-x+1 \text { together with the line } y=x . \text { At what angle do }} \\ \quad {\text { the lines intersect? }} \\ {\text { b. Find the inverse of } f(x)=-x+b(b \text { constant). What angle }} \\ \quad {\text { does the line } y=-x+b \text { make with the line } y=x ?}\\ {\text { c. What can you conclude about the inverses of functions whose }} \\ \quad {\text { graphs are lines perpendicular to the line } y=x ?}\end{array}\end{equation}

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Problem 49

Show that increasing functions and decreasing functions are one-to-one. That is, show that for any $x_{1}$ and $x_{2}$ in $I, x_{2} \neq x_{1}$ implies $f\left(x_{2}\right) \neq f\left(x_{1}\right).$

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Problem 50

Use the results of Exercise 49 to show that the functions in Exercises $50-54$ have inverses over their domains. Find a formula for $d f^{-1} / d x$ using Theorem $1.$
$$f(x)=(1 / 3) x+(5 / 6)$$

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Problem 51

Use the results of Exercise 49 to show that the functions in Exercises $50-54$ have inverses over their domains. Find a formula for $d f^{-1} / d x$ using Theorem $1.$
$$f(x)=27 x^{3}$$

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Problem 52

Use the results of Exercise 49 to show that the functions in Exercises $50-54$ have inverses over their domains. Find a formula for $d f^{-1} / d x$ using Theorem $1.$
$$f(x)=1-8 x^{3}$$

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Problem 53

Use the results of Exercise 49 to show that the functions in Exercises $50-54$ have inverses over their domains. Find a formula for $d f^{-1} / d x$ using Theorem $1.$
$$f(x)=(1-x)^{3}$$

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Problem 54

Use the results of Exercise 49 to show that the functions in Exercises $50-54$ have inverses over their domains. Find a formula for $d f^{-1} / d x$ using Theorem $1.$
$$f(x)=x^{5 / 3}$$

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Problem 55

If $f(x)$ is one-to-one, can anything be said about $g(x)=-f(x) ?$ Is it also one-to-one? Give reasons for your answer.

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Problem 56

If $f(x)$ is one-to-one and $f(x)$ is never zero, can anything be said about $h(x)=1 / f(x) ?$ Is it also one-to-one? Give reasons for your answer.

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Problem 57

Suppose that the range of $g$ lies in the domain of $f$ so that the composite $f \circ g$ is defined. If $f$ and $g$ are one-to-one, can anything be said about $f \circ g ?$ Give reasons for your answer.

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Problem 58

If a composite $f \circ g$ is one-to-one, must $g$ be one-to-one? Give reasons for your answer.

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Problem 59

Assume that $f$ and $g$ are differentiable functions that are inverses of one another so that $(g \circ f)(x)=x .$ Differentiate both sides of this equation with respect to $x$ using the Chain Rule to express $(g \circ f)^{\prime}(x)$ as a product of derivatives of $g$ and $f .$ What do you find? (This is not a proof of Theorem 1 because we assume here the theorem's conclusion that $g=f^{-1}$ is differentiable.)

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Problem 60

Equivalence of the washer and shell methods for finding volume Let $f$ be differentiable and increasing on the interval $a \leq x \leq b,$ with $a>0,$ and suppose that $f$ has a differentiable inverse, $f^{-1}.$ Revolve about the $y$ -axis the region bounded by the graph of $f$ and the lines $x=a$ and $y=f(b)$ to generate a solid. Then the values of the integrals given by the washer and shell methods for the volume have identical values:
$$\int_{f(a)}^{f(b)} \pi\left(f^{-1}(y)\right)^{2}-a^{2} ) d y=\int_{a}^{b} 2 \pi x(f(b)-f(x)) d x$$
To prove this equality, define
$$\begin{aligned} W(t) &=\int_{f(a)}^{f(t)} \pi\left(\left(f^{-1}(y)\right)^{2}-a^{2}\right) d y \\ S(t) &=\int_{a}^{t} 2 \pi x(f(t)-f(x)) d x \end{aligned}$$
Then show that the functions $W$ and $S$ agree at a point of $[a, b]$ and have identical derivatives on $[a, b] .$ As you saw in Section 4.7 Exercise 90 , this will guarantee $W(t)=S(t)$ for all $t$ in $[a, b] .$ In particular, $W(b)=S(b) .$ (Source: "Disks and Shells Revisited," by Walter Carlip, American Mathematical Monthly, Vol. $98,$ No. $2,$
Feb. $1991,$ pp. $154-156 .$ )

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Problem 61

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=\sqrt{3 x-2}, \quad \frac{2}{3} \leq x \leq 4, \quad x_{0}=3$$

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Problem 62

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=\frac{3 x+2}{2 x-11}, \quad-2 \leq x \leq 2, \quad x_{0}=1 / 2$$

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Problem 63

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=\frac{4 x}{x^{2}+1}, \quad-1 \leq x \leq 1, \quad x_{0}=1 / 2$$

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Problem 64

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=\frac{x^{3}}{x^{2}+1}, \quad-1 \leq x \leq 1, \quad x_{0}=1 / 2$$

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Problem 65

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=x^{3}-3 x^{2}-1, \quad 2 \leq x \leq 5, \quad x_{0}=\frac{27}{10}$$

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Problem 66

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=2-x-x^{3}, \quad-2 \leq x \leq 2, \quad x_{0}=\frac{3}{2}$$

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Problem 67

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=e^{x}, \quad-3 \leq x \leq 5, \quad x_{0}=1$$

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Problem 68

You will explore some functions and their inverses together with their derivatives and linear approximating functions at specified points. Perform the following steps using your CAS:
\begin{equation}
\begin{array}{l}{\text { a. Plot the function } y=f(x) \text { together with its derivative over the given }} \\ \quad {\text { interval. Explain why you know that } f \text { is one-to-one over the interval. }} \\ {\text { b. Solve the equation } y=f(x) \text { for } x \text { as a function of } y, \text { and name the }} \\ \quad {\text { resulting inverse function } g \text { . }} \\ {\text { c. Find the equation for the tangent line to } f \text { at the specified point }} \\ {\quad\left(x_{0}, f\left(x_{0}\right)\right) .} \\ {\text { d. Find the equation for the tangent line to } g \text { at the point }\left(f\left(x_{0}\right), x_{0}\right)} \\ \quad {\text { located symmetrically across the } 45^{\circ} \text { line } y=x \text { (which is the }} \\ \quad {\text { graph of the identity function). Use Theorem } 1 \text { to find the slope of }} \\ \quad {\text { this tangent line. }}\\ {\text { e. Plot the functions } f \text { and } g, \text { the identity, the two tangent lines, and }} \\ \quad {\text { the line segment joining the points }\left(x_{0}, f\left(x_{0}\right)\right) \text { and }\left(f\left(x_{0}\right), x_{0}\right) . \text { Discuss }} \\ \quad {\text { the symmetries you see across the main diagonal. }} \end{array}
\end{equation}

$$y=\sin x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \quad x_{0}=1$$

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Problem 69

Repeat the steps above to solve for the functions $y=f(x)$ and $x=f^{-1}(y)$ defined implicitly by the given equations over the interval.
$$y^{1 / 3}-1=(x+2)^{3}, \quad-5 \leq x \leq 5, \quad x_{0}=-3 / 2$$

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Problem 70

Repeat the steps above to solve for the functions $y=f(x)$ and $x=f^{-1}(y)$ defined implicitly by the given equations over the interval.
$$\cos y=x^{1 / 5}, \quad 0 \leq x \leq 1, \quad x_{0}=1 / 2$$

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