Chapter 3

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(x=\sqrt{1-y^{2}}\)

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions. \(f(x)=\sqrt[3]{x-1}\) and \(g(x)=x^{3}+1\)

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph. \(y=|x+1|\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x)-7\)

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=\frac{1}{x-4}, \quad g(x)=\frac{2}{x}+4\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{x-3}{x^{2}+9 x-22}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y=\frac{3 x+5}{7 x-1}\)

For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions. \(f(x)=-3 x+5\) and \(g(x)=\frac{x-5}{-3}\)

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph. \(y=|x|+1\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x-2)+3\)

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. \(f(x)=x^{4}+6, g(x)=x-6\) and \(h(x)=\sqrt{x}\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{1}{x^{2}-x-6}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(x^{2}+y^{2}=9\)

For the following exercises, use a graphing utility to determine whether each function is one-to-one. \(f(x)=\sqrt{x}\)

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

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x+4)-1\)

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. \(f(x)=x^{2}+1, g(x)=\frac{1}{x},\) and \(h(x)=x+3\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{2 x^{3}-250}{x^{2}-2 x-15}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(2 x y=1\)

For the following exercises, use a graphing utility to determine whether each function is one-to-one. \(f(x)=\sqrt[3]{3 x+1}\)

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

For the following exercises, determine the interval(s) on which the function is increasing and decreasing. \(f(x)=4(x+1)^{2}-5\)

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. Given \(f(x)=\frac{1}{x}\) and \(g(x)=x-3,\) find the following: (a) \((f \circ g)(x)\) (b) the domain of \((f \circ g)(x)\) in interval notation (c) \((g \circ f)(x)\) (d) the domain of \((g \circ f)(x)\) (e) \(\left(\frac{f}{g}\right)(x)\)

For the following exercises, find the domain of each function using interval notation. \(\frac{5}{\sqrt{x-3}}\)

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

For the following exercises, use a graphing utility to determine whether each function is one-to-one. \(f(x)=-5 x+1\)

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

For the following exercises, determine the interval(s) on which the function is increasing and decreasing. \(g(x)=5(x+3)^{2}-2\)

For the following exercises, find the domain of each function using interval notation. \(\frac{2 x+1}{\sqrt{5-x}}\)

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

For the following exercises, use a graphing utility to determine whether each function is one-to-one. \(f(x)=x^{3}-27\)

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

For the following exercises, determine the interval(s) on which the function is increasing and decreasing. \(a(x)=\sqrt{-x+4}\)

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. Given the functions \(f(x)=\frac{1-x}{x}\) and \(g(x)=\frac{1}{1+x^{2}},\) find the following: (a) \((g \circ f)(x)\) (b) \((g \circ f)(2)\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{\sqrt{x-4}}{\sqrt{x-6}}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y=\sqrt{1-x^{2}}\)

For the following exercises, graph the given functions by hand. \(f(x)=-|x-1|-2\)

For the following exercises, determine the interval(s) on which the function is increasing and decreasing. \(k(x)=-3 \sqrt{x}-1\)

For the following exercises, use each set of functions to find \(f(g(h(x)))\). Simplify your answers. Given functions \(p(x)=\frac{1}{\sqrt{x}}\) and \(m(x)=x^{2}-4,\) state the domain of each of the following functions using interval notation: (a) \(\frac{p(x)}{m(x)}\) (b) \(p(m(x))\) (c) \(m(p(x))\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{\sqrt{x-6}}{\sqrt{x-4}}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(x=\pm \sqrt{1-y}\)

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

Given functions \(q(x)=\frac{1}{\sqrt{x}}\) and \(h(x)=x^{2}-9,\) state the domain of each of the following functions using interval notation. (a) \(\frac{q(x)}{h(x)}\) (b) \(q(h(x))\) (c) \(h(q(x))\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{x}{x}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y=\pm \sqrt{1-x}\)

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

For \(f(x)=\frac{1}{x}\) and \(g(x)=\sqrt{x-1},\) write the domain of \((f \circ g)(x)\) in interval notation.

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{x^{2}-9 x}{x^{2}-81}\)

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

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