Chapter 3

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=3 x^{2}\) and \(g(x)=\sqrt{x-5},\) find \(f+g, \quad f-g, \quad f g,\) and \(\frac{f}{g}\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(f(x)=2 x^{2}+1\) on \([x, x+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=3-\sqrt{6-2 x}\)

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

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=3-x\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-3|x-2|-1\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x-49)\)

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=\sqrt{x}\) and \(g(x)=|x-3|,\) find \(\frac{g}{f}\).

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(g(x)=3 x^{2}-2\) on \([x, x+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\sqrt{4-3 x}\)

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

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=\frac{x}{x+2}\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-2|x+1|+6\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x+43)\)

For the following exercise, find the indicated function given \(f(x)=2 x^{2}+1\) and \(g(x)=3 x-5\). (a) \(f(g(2))\) (b) \(f(g(x))\) (c) \(g(f(x))\) (d) \((g \circ g)(x)\) (e) \((f \circ f)(-2)\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(a(t)=\frac{1}{t+4}\) on \([9,9+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\sqrt{x^{2}+4}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(3 x^{2}+y=14\)

For the following exercises, find \(f^{-1}(x)\) for each function. \(f(x)=\frac{2 x+3}{5 x+4}\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-5|x+2|+15\)

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

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

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(b(x)=\frac{1}{x+3}\) on \([1,1+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\sqrt[3]{1-2 x}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(2 x+y^{2}=6\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=2|x-1|-6\)

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

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

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(j(x)=3 x^{3}\) on \([1,1+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\sqrt[3]{x-1}\)

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y=-2 x^{2}+40 x\)

For the following exercises, find a domain on which each function \(f\) is one- to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain. \(f(x)=(x-6)^{2}\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=|-2 x+1|-13\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x)+5\)

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=|x|, \quad g(x)=5 x+1\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(r(t)=4 t^{3}\) on \([2,2+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{9}{x-6}\)

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

For the following exercises, find a domain on which each function \(f\) is one- to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain. \(f(x)=x^{2}-5\)

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-|x-9|+16\)

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x)+8\)

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=\sqrt[3]{x}, \quad g(x)=\frac{x+1}{x^{3}}\)

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. \(\frac{f(x+h)-f(x)}{h}\) given \(f(x)=2 x^{2}-3 x\) on \([x, x+h]\)

For the following exercises, find the domain of each function using interval notation. \(f(x)=\frac{3 x+1}{4 x+2}\)

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

Given \(f(x)=\frac{x}{2+x}\) and \(g(x)=\frac{2 x}{1-x}:\) (a) Find \(f(g(x))\) and \(g(f(x))\). (b) What does the answer tell us about the relationship between \(f(x)\) and \(g(x) ?\)

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 \(f\). \(y=f(x)-2\)

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x))\). Simplify your answers. \(f(x)=\frac{1}{x-6}, \quad g(x)=\frac{7}{x}+6\)

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