Classify each of the following statements as either true or false.

The composition of two functions $f$ and $g$ is written $f \circ g$.

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Classify each of the following statements as either true or false.

The notation $(f \circ g)(x)$ means $f(g(x))$.

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Classify each of the following statements as either true or false.

If $f(x)=x^{2}$ and $g(x)=x+3,$ then $(g \circ f)(x)=(x+3)^{2}$.

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Classify each of the following statements as either true or false.

For any function $h,$ there is only one way to decompose the function as $h=f \circ g$.

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Classify each of the following statements as either true or false.

The function $f$ is one-to-one if $f(1)=1$.

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Classify each of the following statements as either true or false.

The $-1$ in $f^{-1}$ is an exponent.

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Classify each of the following statements as either true or false.

The function $f$ is the inverse of $f^{-1}$.

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Classify each of the following statements as either true or false.

If $g$ and $h$ are inverses of each other, then $(g \circ h)(x)=x$.

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=x^{2}+1 ; g(x)=x-3$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=x+4 ; g(x)=x^{2}-5$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=5 x+1 ; g(x)=2 x^{2}-7$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=3 x^{2}+4 ; g(x)=4 x-1$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=x+7 ; g(x)=1 / x^{2}$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=1 / x^{2} ; g(x)=x+2$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=\sqrt{x ;} g(x)=x+3$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=10-x ; g(x)=\sqrt{x}$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=\sqrt{4 x} ; g(x)=1 / x$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=\sqrt{x+3} ; g(x)=13 / x$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=x^{2}+4 ; g(x)=\sqrt{x-1}$$

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For each pair of functions, find (a) $(f \circ g)(1)$ (b) $(g \circ f)(1) ;(\mathbf{c})(f \circ g)(x) ;$ and $(\mathbf{d})(g \circ f)(x)$.

$$f(x)=x^{2}+8 ; g(x)=\sqrt{x+17}$$

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Use the following table to find each value, if possible.

$$\begin{array}{|c|c|c|}\hline X & {Y 1} & {Y_{2}} \\

\hline-3 & {-4} & {1} \\

{-2} & {-1} & {-2} \\

{-1} & {2} & {-3} \\

{0} & {5} & {-2} \\

{1} & {8} & {1} \\

{2} & {11} & {6} \\

{3} & {14} & {11} \\

\hline {\text {X} =} \\ \hline \end{array}$$

$$\left(y_{1} \circ y_{2}\right)(-3)$$

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Use the following table to find each value, if possible.

$$\begin{array}{|c|c|c|}\hline X & {Y 1} & {Y_{2}} \\

\hline-3 & {-4} & {1} \\

{-2} & {-1} & {-2} \\

{-1} & {2} & {-3} \\

{0} & {5} & {-2} \\

{1} & {8} & {1} \\

{2} & {11} & {6} \\

{3} & {14} & {11} \\

\hline {\text {X} =} \\ \hline \end{array}$$

$$\left(y_{2} \circ y_{1}\right)(-3)$$

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Use the following table to find each value, if possible.

$$\begin{array}{|c|c|c|}\hline X & {Y 1} & {Y_{2}} \\

\hline-3 & {-4} & {1} \\

{-2} & {-1} & {-2} \\

{-1} & {2} & {-3} \\

{0} & {5} & {-2} \\

{1} & {8} & {1} \\

{2} & {11} & {6} \\

{3} & {14} & {11} \\

\hline {\text {X} =} \\ \hline \end{array}$$

$$\left(y_{1} \circ y_{2}\right)(-1)$$

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Use the following table to find each value, if possible.

$$\begin{array}{|c|c|c|}\hline X & {Y 1} & {Y_{2}} \\

\hline-3 & {-4} & {1} \\

{-2} & {-1} & {-2} \\

{-1} & {2} & {-3} \\

{0} & {5} & {-2} \\

{1} & {8} & {1} \\

{2} & {11} & {6} \\

{3} & {14} & {11} \\

\hline {\text {X} =} \\ \hline \end{array}$$

$$\left(y_{2} \circ y_{1}\right)(-1)$$

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Use the following table to find each value, if possible.

$$\begin{array}{|c|c|c|}\hline X & {Y 1} & {Y_{2}} \\

\hline-3 & {-4} & {1} \\

{-2} & {-1} & {-2} \\

{-1} & {2} & {-3} \\

{0} & {5} & {-2} \\

{1} & {8} & {1} \\

{2} & {11} & {6} \\

{3} & {14} & {11} \\

\hline {\text {X} =} \\ \hline \end{array}$$

$$\left(y_{2} \circ y_{1}\right)(1)$$

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Use the following table to find each value, if possible.

$$\begin{array}{|c|c|c|}\hline X & {Y 1} & {Y_{2}} \\

\hline-3 & {-4} & {1} \\

{-2} & {-1} & {-2} \\

{-1} & {2} & {-3} \\

{0} & {5} & {-2} \\

{1} & {8} & {1} \\

{2} & {11} & {6} \\

{3} & {14} & {11} \\

\hline {\text {X} =} \\ \hline \end{array}$$

$$\left(y_{1} \circ y_{2}\right)(1)$$

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Use the table below to find each value, if possible.

$$\begin{array}{|c|c|c|} \hline

{x} & {f(x)} & {g(x)} \\

\hline {1} & {0} & {1} \\

{2} & {3} & {5} \\

{3} & {2} & {8} \\

{4} & {6} & {5} \\

{5} & {4} & {1} \\ \hline

\end{array}$$

$$(f \circ g)(2)$$

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Use the table below to find each value, if possible.

$$\begin{array}{|c|c|c|} \hline

{x} & {f(x)} & {g(x)} \\

\hline {1} & {0} & {1} \\

{2} & {3} & {5} \\

{3} & {2} & {8} \\

{4} & {6} & {5} \\

{5} & {4} & {1} \\ \hline

\end{array}$$

$$(g \circ f)(4)$$

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Use the table below to find each value, if possible.

$$\begin{array}{|c|c|c|} \hline

{x} & {f(x)} & {g(x)} \\

\hline {1} & {0} & {1} \\

{2} & {3} & {5} \\

{3} & {2} & {8} \\

{4} & {6} & {5} \\

{5} & {4} & {1} \\ \hline

\end{array}$$

$$f(g(3))$$

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Use the table below to find each value, if possible.

$$\begin{array}{|c|c|c|} \hline

{x} & {f(x)} & {g(x)} \\

\hline {1} & {0} & {1} \\

{2} & {3} & {5} \\

{3} & {2} & {8} \\

{4} & {6} & {5} \\

{5} & {4} & {1} \\ \hline

\end{array}$$

$$g(f(5))$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=(3 x-5)^{4}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=(2 x+7)^{3}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\sqrt{2 x+7}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\sqrt[3]{4 x-5}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\frac{2}{x-3}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\frac{3}{x}+4$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\frac{1}{\sqrt{7 x+2}}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\sqrt{x-7}-3$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\frac{1}{\sqrt{3 x}}+\sqrt{3 x}$$

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Find $f(x)$ and $g(x)$ such that $h(x)=(f \circ g)(x)$. Answers may vary.

$$h(x)=\frac{1}{\sqrt{2 x}}-\sqrt{2 x}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=x+4$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=x+2$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=2 x$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=3 x$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$g(x)=3 x-1$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$g(x)=2 x-5$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{1}{2} x+1$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{1}{3} x+2$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$g(x)=x^{2}+5$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$g(x)=x^{2}-4$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$h(x)=-10-x$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$h(x)=7-x$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{1}{x}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{3}{x}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$G(x)=4$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$H(x)=2$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{2 x+1}{3}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\frac{3 x+2}{5}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=x^{3}-5$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=x^{3}+7$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$g(x)=(x-2)^{3}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$g(x)=(x+7)^{3}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\sqrt{x}$$

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For each function, (a) determine whether it is one-to-one and (b) if it is one-to-one, find a formula for the inverse.

$$f(x)=\sqrt{x-1}$$

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Graph each function and its inverse using the same set of axes.

$$f(x)=\frac{2}{3} x+4$$

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Graph each function and its inverse using the same set of axes.

$$g(x)=\frac{1}{4} x+2$$

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Graph each function and its inverse using the same set of axes.

$$f(x)=x^{3}+1$$

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Graph each function and its inverse using the same set of axes.

$$f(x)=x^{3}-1$$

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Graph each function and its inverse using the same set of axes.

$$g(x)=\frac{1}{2} x^{3}$$

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Graph each function and its inverse using the same set of axes.

$$g(x)=\frac{1}{3} x^{3}$$

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Graph each function and its inverse using the same set of axes.

$$F(x)=-\sqrt{x}$$

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Graph each function and its inverse using the same set of axes.

$$f(x)=\sqrt{x}$$

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Graph each function and its inverse using the same set of axes.

$$f(x)=-x^{2}, x \geq 0$$

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Graph each function and its inverse using the same set of axes.

$$f(x)=x^{2}-1, x \leq 0$$

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Let $f(x)=\sqrt[3]{x-4} .$ Use composition to show that $f^{-1}(x)=x^{3}+4$.

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Let $f(x)=3 /(x+2) .$ Use composition to show that $f^{-1}(x)=\frac{3}{x}-2$.

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Let $f(x)=(1-x) / x .$ Use composition to show that $f^{-1}(x)=\frac{1}{x+1}$.

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Let $f(x)=x^{3}-5 .$ Use composition to show that $f^{-1}(x)=\sqrt[3]{x+5}$.

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Use a graphing calculator to help determine whether or not the given pairs of functions are inverses of each other.

$$f(x)=0.75 x^{2}+2 ; g(x)=\sqrt{\frac{4(x-2)}{3}}$$

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Use a graphing calculator to help determine whether or not the given pairs of functions are inverses of each other.

$$f(x)=1.4 x^{3}+3.2 ; g(x)=\sqrt[3]{\frac{x-3.2}{1.4}}$$

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Use a graphing calculator to help determine whether or not the given pairs of functions are inverses of each other.

$$\begin{aligned}

&f(x)=\sqrt{2.5 x+9.25}\\

&g(x)=0.4 x^{2}-3.7, x \geq 0

\end{aligned}$$

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Use a graphing calculator to help determine whether or not the given pairs of functions are inverses of each other.

$$\begin{array}{l}

{f(x)=0.8 x^{1 / 2}+5.23} \\

{g(x)=1.25\left(x^{2}-5.23\right), x \geq 0}

\end{array}$$

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In Exercises 91 and 92, match the graph of each function in Column A with the graph of its inverse in Column B.

GRAPH CAN'T COPY

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GRAPH CAN'T COPY

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Dress Sizes in the United States and France. A size-6 dress in the United States is size 38 in France. A function that converts dress sizes in the United States to those in France is

$$f(x)=x+32.$$

a) Find the dress sizes in France that correspond to sizes $8,10,14,$ and 18 in the United States.

b) Determine whether this function has an inverse that is a function. If so, find a formula for the inverse.

c) Use the inverse function to find dress sizes in the United States that correspond to sizes $40,42,46$ and 50 in France.

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Dress Sizes in the United States and Italy. A size-6 dress in the United States is size 36 in Italy. A function that converts dress sizes in the United States to those in Italy is

$$f(x)=2(x+12).$$

a) Find the dress sizes in Italy that correspond to sizes $8,10,14,$ and 18 in the United States.

b) Determine whether this function has an inverse that is a function. If so, find a formula for the inverse.

c) Use the inverse function to find dress sizes in the United States that correspond to sizes $40,44,52$ and 60 in Italy.

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Is there a one-to-one relationship between items in a store and the price of each of those items? Why or why not?

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Mathematicians usually try to select "logical" words when forming definitions. Does the term "one-to-one" seem logical? Why or why not?

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To prepare for Section 12.2, review simplifying exponential expressions and graphing equations (Sections 3.2, 5.2, and 10.2).

Simplify.

$$2^{-3}[5.2]$$

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To prepare for Section 12.2, review simplifying exponential expressions and graphing equations (Sections 3.2, 5.2, and 10.2).

Simplify.

$$5^{(1-3)}[5.2]$$

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To prepare for Section 12.2, review simplifying exponential expressions and graphing equations (Sections 3.2, 5.2, and 10.2).

Simplify.

$$4^{5 / 2}[10.2]$$

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To prepare for Section 12.2, review simplifying exponential expressions and graphing equations (Sections 3.2, 5.2, and 10.2).

Simplify.

$$3^{7 / 10}[10.2]$$

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The function $V(t)=750(1.2)^{t}$ is used to predict the value, $V(t),$ of a certain rare stamp $t$ years from $2005 .$ Do not calculate $V^{-1}(t),$ but explain how $V^{-1}$ could be used.

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An organization determines that the cost per person of chartering a bus is given by the function

$$C(x)=\frac{100+5 x}{x}$$

where $x$ is the number of people in the group and $C(x)$ is in dollars. Determine $C^{-1}(x)$ and explain how this inverse function could be used.

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Dress Sizes in France and ltaly. Use the information in Exercises 93 and 94 to find a function for the French dress size that corresponds to a size $x$ dress in Italy.

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Dress Sizes in Italy and France. Use the information in Exercises 93 and 94 to find a function for the Italian dress size that corresponds to a size $x$ dress in France.

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What relationship exists between the answers to Exercises 107 and 108? Explain how you determined this.

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Show that function composition is associative by showing that $((f \circ g) \circ h)(x)=(f \circ(g \circ h))(x)$.

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Show that if $h(x)=(f \circ g)(x),$ then $h^{-1}(x)=$ $\left(g^{-1} \circ f^{-1}\right)(x) \cdot(\text {Hint}: \text { Use Exercise } 110 .)$

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Match each function in Column A with its inverse from Column B.

Column A

(1) $y=5 x^{3}+10$

(2) $y=(5 x+10)^{3}$

(3) $y=5(x+10)^{3}$

(4) $y=(5 x)^{3}+10$

Column B

A. $y=\frac{\sqrt[3]{x}-10}{5}$

B. $y=\sqrt[3]{\frac{x}{5}}-10$

C. $y=\sqrt[3]{\frac{x-10}{5}}$

D. $y=\frac{\sqrt[3]{x-10}}{5}$

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Examine the following table. Is it possible that $f$ and $g$ could be inverses of each other? Why or why not?

$$\begin{array}{|c|c|c|} \hline

{x} & {f(x)} & {g(x)} \\

{6} & {6} & {6} \\

{7} & {6.5} & {8} \\

{8} & {7} & {10} \\

{9} & {7.5} & {12} \\

{10} & {8} & {14} \\

{11} & {8.5} & {16} \\

{12} & {9} & {18} \\ \hline

\end{array}$$

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Assume in Exercise 113 that $f$ and $g$ are both linear functions. Find equations for $f(x)$ and $g(x)$ Are $f$ and $g$ inverses of each other?

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Let $c(w)$ represent the cost of mailing a package that weighs $w$ pounds. Let $f(n)$ represent the weight, in pounds, of $n$ copies of a certain book. Explain what $(c \circ f)(n)$ represents.

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Let $g(a)$ represent the number of gallons of sealant needed to seal a bamboo floor with area $a .$ Let $c(s)$ represent the cost of $s$ gallons of sealant. Which composition makes sense: $(c \circ g)(a)$ or $(g \circ c)(s) ?$ What does it represent?

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The following graphs show the rate of flow $R,$ in liters per minute, of blood from the heart in a man who bicycles for 20 min, and the pressure $P,$ in millimeters of mercury, in the artery leading to the lungs for a rate of blood flow $R$ from the heart. (This problem was suggested by Kandace Kling, Portland Community College, Sylvania, Oregon.

GRAPH CAN'T COPY

Estimate $P(R(10))$.

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The following graphs show the rate of flow $R,$ in liters per minute, of blood from the heart in a man who bicycles for 20 min, and the pressure $P,$ in millimeters of mercury, in the artery leading to the lungs for a rate of blood flow $R$ from the heart. (This problem was suggested by Kandace Kling, Portland Community College, Sylvania, Oregon.

GRAPH CAN'T COPY

Explain what $P(R(10))$ represents.

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The following graphs show the rate of flow $R,$ in liters per minute, of blood from the heart in a man who bicycles for 20 min, and the pressure $P,$ in millimeters of mercury, in the artery leading to the lungs for a rate of blood flow $R$ from the heart. (This problem was suggested by Kandace Kling, Portland Community College, Sylvania, Oregon.

GRAPH CAN'T COPY

Estimate $P^{-1}(20)$.

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