Chapter 3

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=(x+2)^{2}\)

Find the domain of the function \(f(x)=\sqrt{2 x^{3}-50 x}\) by: (a) using algebra. (b) graphing the function in the radicand and determining intervals on the \(x\) -axis for which the radicand is nonnegative.

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y^{3}=x^{2}\)

For the following exercises, graph the given functions by hand. \(f(x)=|2 x-4|-3\)

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. \(f(t)=(t+1)^{2}-3\)

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=(x-5)^{3}\)

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=2 x-5 \quad\)

For the following exercises, graph the given functions by hand. \(f(x)=|3 x+9|+2\)

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. \(h(x)=|x-1|+4\)

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\frac{3}{x-5}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(f(x)=x^{2}\) on [1,5]

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=-5 x^{2}+2 x-1\)

For the following exercises, graph the given functions by hand. \(f(x)=-|x-1|-3\)

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. \(k(x)=(x-2)^{3}-1\)

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\frac{4}{(x+2)^{2}}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(h(x)=5-2 x^{2}\) on [-2,4]

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=\sqrt{2-x}+5\)

For the following exercises, graph the given functions by hand. \(f(x)=-|x+4|-3\)

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. \(m(t)=3+\sqrt{t+2}\)

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=4+\sqrt[3]{x}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(q(x)=x^{3}\) on [-4,2]

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=\frac{6 x-1}{5 x+2}\)

For the following exercises, graph the given functions by hand. \(f(x)=\frac{1}{2}|x+4|-3\)

Tabular representations for the functions \(f, \quad g,\) and \(h\) are given below. Write \(g(x)\) and \(h(x)\) as transformations of \(f(x)\).

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\sqrt[3]{\frac{1}{2 x-3}}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(g(x)=3 x^{3}-1\) on [-3,3]

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=|x-1|-|x+1|\)

Use a graphing utility to graph \(f(x)=10|x-2|\) on the viewing window [0,4] . Identify the corresponding range. Show the graph.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\frac{1}{\left(3 x^{2}-4\right)^{-3}}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(y=\frac{1}{x}\) on [1,3]

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f(6)=7,\) find \(f^{-1}(7)\)

Use a graphing utility to graph \(f(x)=-100|x|+100\) on the viewing window [-5,5] . Identify the corresponding range. Show the graph.

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\sqrt[4]{\frac{3 x-2}{x+5}}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(p(t)=\frac{\left(t^{2}-4\right)(t+1)}{t^{2}+3}\) on [-3,1]

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f(3)=2,\) find \(f^{-1}(2)\)

For the following exercises, graph each function using a graphing utility. Specify the viewing window. \(f(x)=-0.1|0.1(0.2-x)|+0.3\)

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\left(\frac{8+x^{3}}{8-x^{3}}\right)^{4}\)

For the following exercises, find the average rate of change of each function on the interval specified. \(k(t)=6 t^{2}+\frac{4}{t^{3}}\) on [-1,3]

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(k(t)=2 t-1:\) (a) Evaluate \(k(2)\). (b) Solve \(k(t)=7\).

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f^{-1}(-4)=-8\), find \(f(-8)\).

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\sqrt{2 x+6}\)

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(f(x)=x^{4}-4 x^{3}+5\)

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(f(x)=8-3 x:\) (a) Evaluate \(f(-2)\) (b) Solve \(f(x)=-1\).

For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one. If \(f^{-1}(-2)=-1,\) find \(f(-1)\)

If possible, find all values of \(a\) such that there are no \(x\) - intercepts for \(f(x)=2|x+1|+a\).

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=(5 x-1)^{3}\)

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(h(x)=x^{5}+5 x^{4}+10 x^{3}+10 x^{2}-1\)

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(p(c)=c^{2}+c:\) (a) Evaluate \(p(-3)\). (b) Solve \(p(c)=2\).

For the following exercises, use the values listed in to evaluate or solve. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline f(x) & 8 & 0 & 7 & 4 & 2 & 6 & 5 & 3 & 9 & 1 \\ \hline \end{array} $$ Find \(f(1)\).

If possible, find all values of \(a\) such that there are no \(y\) -intercepts for \(f(x)=2|x+1|+a\).