College Algebra

Michael Sullivan

Chapter 6

Exponential and Logarithmic Functions

Educators


Problem 1

\text { Find } f(3) \text { if } f(x)=-4 x^{2}+5 x .(\mathrm{pp} .203-206)

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Problem 2

\text { Find } f(3 x) \text { if } f(x)=4-2 x^{2},(p p, 203-206)

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Problem 3

Find the domain of the function $f(x)=\frac{x^{2}-1}{x^{2}-25}$ $(p p .206-208)$

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Problem 4

Given two functions $f$ and $g,$ _____ _____ the denoted $f \circ g,$ is defined by $f \circ g(x)=$ _____.

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Problem 5

True or False $f(g(x))=f(x) \cdot g(x)$.

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Problem 6

True or False The domain of the composite function $(f \circ g)(x)$ is the same as the domain of $g(x)$.

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Problem 7

Evaluate each expression using the values given in the table.
$$\begin{array}{|l|rrrrrrr|}\hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\f(x) & -7 & -5 & -3 & -1 & 3 & 5 & 7 \\g(x) & 8 & 3 & 0 & -1 & 0 & 3 & 8 \\\hline\end{array}$$
(a) $(f \circ g)(1)$
(b) $(f \circ g)(-1)$
(c) $(g \circ f)(-1)$
(d) $(g \circ f)(0)$
(c) $(g \circ g)(-2)$
(f) $(f \circ f)(-1)$

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Problem 8

Evaluate each expression using the values given in the table.
$$\begin{array}{|l|lllllll|}\hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\f(x) & 11 & 9 & 7 & 5 & 3 & 1 & -1 \\g(x) & -8 & -3 & 0 & 1 & 0 & -3 & -8 \\\hline\end{array}$$
(a) $(f \circ g)(1)$
(b) $(f \circ g)(2)$
(c) $(g \circ f)(2)$
(d) $(g \circ f)(3)$
(e) $(g \circ g)(1)$
(f) $(f \circ f)(3)$

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Problem 9

Evaluate each expression using the graphs of $y=f(x)$ and $y=g(x)$ shown in the figure.
(a) $(g \circ f)(-1)$
(b) $(g \circ f)(0)$
(c) $(f \circ g)(-1)$
(d) $(f \circ g)(4)$
CAN'T COPY THE GRAPH

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Problem 10

Evaluate each expression using the graphs of $y=f(x)$ and $y=g(x)$ shown in the figure.
(a) $(g \circ f)(1)$
(b) $(g \circ f)(5)$
(c) $(f \circ g)(0)$
(d) $(f \circ g)(2)$
CAN'T COPY THE GRAPH

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Problem 11

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=2 x ; \quad g(x)=3 x^{2}+1$$

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Problem 12

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=3 x+2 ; \quad g(x)=2 x^{2}-1$$

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Problem 13

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=4 x^{2}-3 ; \quad g(x)=3-\frac{1}{2} x^{2}$$

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Problem 14

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=2 x^{2} ; \quad g(x)=1-3 x^{2}$$

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Problem 15

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=\sqrt{x} ; \quad g(x)=2 x$$

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Problem 16

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=\sqrt{x+1} ; \quad g(x)=3 x$$

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Problem 17

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=|x| ; \quad g(x)=\frac{1}{x^{2}+1}$$

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Problem 18

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=|x-2| ; \quad g(x)=\frac{3}{x^{2}+2}$$

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Problem 19

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=\frac{3}{x+1} ; \quad g(x)=\sqrt[3]{x}$$

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Problem 20

For the given functions $f$ and $g,$ find:
$$\left(\begin{array}{llll}a) & (f \circ g)(4)\end{array}\right.$$
$$(b) (g \circ f)(2)$$
$$(c)(f \circ f)(1)$$
$$\left(\begin{array}{llll}d) & (g \circ g)(0)\end{array}\right.$$
$$f(x)=x^{3 / 2} ; \quad g(x)=\frac{2}{x+1}$$

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Problem 21

Find the domain of the composite function $f \circ g .$
$$f(x)=\frac{3}{x-1} ; \quad g(x)=\frac{2}{x}$$

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Problem 22

Find the domain of the composite function $f \circ g .$
$$f(x)=\frac{1}{x+3} ; \quad g(x)=-\frac{2}{x}$$

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Problem 23

Find the domain of the composite function $f \circ g .$
$$f(x)=\frac{x}{x-1} ; \quad g(x)=-\frac{4}{x}$$

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Problem 24

Find the domain of the composite function $f \circ g .$
$$f(x)=\frac{x}{x+3} ; \quad g(x)=\frac{2}{x}$$

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Problem 25

Find the domain of the composite function $f \circ g .$
$$f(x)=\sqrt{x} ; \quad g(x)=2 x+3$$

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Problem 26

Find the domain of the composite function $f \circ g .$
$$f(x)=x-2 ; \quad g(x)=\sqrt{1-x}$$

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Problem 27

Find the domain of the composite function $f \circ g .$
$$f(x)=x^{2}+1 ; \quad g(x)=\sqrt{x-1}$$

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Problem 28

Find the domain of the composite function $f \circ g .$
$$f(x)=x^{2}+4 ; \quad g(x)=\sqrt{x-2}$$

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Problem 29

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=2 x+3 ; \quad g(x)=3 x$$

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Problem 30

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=-x ; \quad g(x)=2 x-4$$

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Problem 31

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=3 x+1 ; \quad g(x)=x^{2}$$

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Problem 32

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=x+1 ; \quad g(x)=x^{2}+4$$

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Problem 33

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=x^{2} ; \quad g(x)=x^{2}+4$$

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Problem 34

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=x^{2}+1 ; \quad g(x)=2 x^{2}+3$$

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Problem 35

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{3}{x-1} ; \quad g(x)=\frac{2}{x}$$

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Problem 36

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{1}{x+3} ; \quad g(x)=-\frac{2}{x}$$

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Problem 37

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{x}{x-1} ; \quad g(x)=-\frac{4}{x}$$

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Problem 38

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{x}{x+3} ; \quad g(x)=\frac{2}{x}$$

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Problem 39

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\sqrt{x} ; \quad g(x)=2 x+3$$

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Problem 40

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\sqrt{x-2} ; \quad g(x)=1-2 x$$

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Problem 41

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=x^{2}+1 ; \quad g(x)=\sqrt{x-1}$$

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Problem 42

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=x^{2}+4 ; \quad g(x)=\sqrt{x-2}$$

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Problem 43

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{x-5}{x+1} ; \quad g(x)=\frac{x+2}{x-3}$$

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Problem 44

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{2 x-1}{x-2} ; \quad g(x)=\frac{x+4}{2 x-5}$$

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Problem 45

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=2 x ; \quad g(x)=\frac{1}{2} x$$

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Problem 46

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=4 x ; \quad g(x)=\frac{1}{4} x$$

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Problem 47

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=x^{3} ; \quad g(x)=\sqrt[3]{x}$$

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Problem 48

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=x+5 ; \quad g(x)=x-5$$

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Problem 49

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=2 x-6 ; \quad g(x)=\frac{1}{2}(x+6)$$

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Problem 50

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=4-3 x ; \quad g(x)=\frac{1}{3}(4-x)$$

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Problem 51

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=a x+b ; \quad g(x)=\frac{1}{a}(x-b) \quad a \neq 0$$

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Problem 52

Show that $(f \circ g)(x)=(g \circ f)(x)=x$.
$$f(x)=\frac{1}{x} ; \quad g(x)=\frac{1}{x}$$

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Problem 53

Find functions $f$ and $g$ so that $f \circ g=H$.
$$H(x)=(2 x+3)^{4}$$

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Problem 54

Find functions $f$ and $g$ so that $f \circ g=H$.
$$H(x)=\left(1+x^{2}\right)^{3}$$

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Problem 55

Find functions $f$ and $g$ so that $f \circ g=H$.
$$H(x)=\sqrt{x^{2}+1}$$

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Problem 56

Find functions $f$ and $g$ so that $f \circ g=H$.
$$H(x)=\sqrt{1-x^{2}}$$

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Problem 57

Find functions $f$ and $g$ so that $f \circ g=H$.
$$H(x)=|2 x+1|$$

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Problem 58

Find functions $f$ and $g$ so that $f \circ g=H$.
$$H(x)=\left|2 x^{2}+3\right|$$

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Problem 59

If $f(x)=2 x^{3}-3 x^{2}+4 x-1$ and $g(x)=2,$ find $(f \circ g)(x)$ and $(g \circ f)(x)$.

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Problem 60

If $f(x)=\frac{x+1}{x-1},$ find $(f \circ f)(x)$.

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Problem 61

If $f(x)=2 x^{2}+5$ and $g(x)=3 x+a,$ find $a$ so that the graph of $f \circ g$ crosses the $y$ -axis at 23.

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Problem 62

If $f(x)=3 x^{2}-7$ and $g(x)=2 x+a,$ find $a$ so that the graph of $f \circ g$ crosses the $y$ -axis at 68.

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Problem 63

Use the functions f and g to find:
$(a) f \circ g$
$(b) g \circ f$
(c) the domain of $f$ o $g$ and of $g \circ f$
(d) the conditions for which $f$ o $g=g \circ f$
$$f(x)=a x+b ; \quad g(x)=c x+d$$

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Problem 64

Use the functions f and g to find:
$(a) f \circ g$
$(b) g \circ f$
(c) the domain of $f$ o $g$ and of $g \circ f$
(d) the conditions for which $f$ o $g=g \circ f$
$$f(x)=\frac{a x+b}{c x+d} ; \quad g(x)=m x$$

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Problem 65

The surface area $S$ (in square meters) of a hot-air balloon is given by
$$S(r)=4 \pi r^{2}$$
where $r$ is the radius of the balloon (in meters). If the radius
$r$ is increasing with time $t$ (in seconds) according to the formula $r(t)=\frac{2}{3} t^{3}, t \geq 0,$ find the surface area $S$ of the balloon as a function of the time $t$

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Problem 66

The volume $V$ (in cubic meters) of the hot-air balloon described in Problem 65 is given by $V(r)=\frac{4}{3} \pi r^{3} .$ If the radius $r$ is the same function of $t$ as in Problem $65,$ find the volume $V$ as a function of the time $t$

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Problem 67

The number $N$ of cars produced at a certain factory in one day after $t$ hours of operation is given by $N(t)=100 t-5 t^{2}, 0 \leq t \leq 10 .$ If the cost $C$ (in dollars) of producing $N$ cars is $C(N)=15,000+8000 N$ find the cost $C$ as a function of the time $t$ of operation of the factory.

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Problem 68

The spread of oil leaking from a tanker is in the shape of a circle. If the radius $r$ (in feet) of the spread after $t$ hours is $r(t)=200 \sqrt{t},$ find the area $A$ of the oil slick as a function of the time $t$.

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Problem 69

The price $p,$ in dollars, of a certain product and the quantity $x$ sold obey the demand equation
$$p=-\frac{1}{4} x+100 \quad 0 \leq x \leq 400$$
Suppose that the cost $C$, in dollars, of producing $x$ units is
$$C=\frac{\sqrt{x}}{25}+600$$
Assuming that all items produced are sold, find the cost $C$ as a function of the price $p$ [Hint: Solve for $x$ in the demand equation and then form the composite. $]$

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Problem 70

cost of a Commodity The price $p,$ in dollars, of a certain commodity and the quantity $x$ sold obey the demand equation
$$p=-\frac{1}{5} x+200 \quad 0 \leq x \leq 1000$$
Suppose that the cost $C,$ in dollars, of producing $x$ units is
$$C=\frac{\sqrt{x}}{10}+400$$
Assuming that all items produced are sold, find the cost $C$ as a function of the price $p$.

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Problem 71

The volume $V$ of a right circular cylinder of height $h$ and radius $r$ is $V=\pi r^{2} h .$ If the height is twice the radius, express the volume $V$ as a function of $r .$

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Problem 72

The volume $V$ of a right circular cone is $V=\frac{1}{3} \pi r^{2} h .$ If the height is twice the radius, express the volume $V$ as a function of $r .$

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Problem 73

Traders often buy foreign currency in hope of making money when the currency's value changes. For example, on June $5,2009,$ one U.S. dollar could purchase 0.7143 Euros, and one Euro could purchase 137.402 yen. Let $f(x)$ represent the number of Euros you can buy with $x$ dollars, and let $g(x)$ represent the number of yen you can buy with $x$ Euros.
(a) Find a function that relates dollars to Euros.
(b) Find a function that relates Euros to yen.
(c) Use the results of parts (a) and (b) to find a function that relates dollars to yen. That is, find $(g \circ f)(x)=$ $g(f(x))$
(d) What is $g(f(1000)) ?$

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Problem 74

The function $C(F)=\frac{5}{9}(F-32)$ converts a temperature in degrees Fahrenheit, $F,$ to a temperature in degrees Celsius, $C$. The function $K(C)=C+273$ converts a temperature in degrees Celsius to a temperature in kelvins, $K$
(a) Find a function that converts a temperature in degrees Fahrenheit to a temperature in kelvins.
(b) Determine 80 degrees Fahrenheit in kelvins.

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Problem 75

The manufacturer of a computer is offering two discounts on last year's model computer. The first discount is a $\$ 200$ rebate and the second discount is $20 \%$ off the regular price, $p$
(a) Write a function $f$ that represents the sale price if only the rebate applies.
(b) Write a function $g$ that represents the sale price if only the $20 \%$ discount applies.
(c) Find $f \circ g$ and $g \circ f .$ What does each of these functions represent? Which combination of discounts represents a better deal for the consumer? Why?

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Problem 76

If $f$ and $g$ are odd functions, show that the composite function $f \circ g$ is also odd.

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Problem 77

If $f$ is an odd function and $g$ is an even function, show that the composite functions $f \circ g$ and $g \circ f$ are both even.

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