91Ó°ÊÓ

The function \(f\) is one-to-one. Find its inverse, and check your answer. State the domain and range of both \(f\) and \(f^{-1}\) $$f(x)=\sqrt{x^{2}-1}, x \geq 1$$

In Exercises \(53-76\), graph the piccewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant. $$f(x)=\left\\{\begin{array}{ll} x^{2} & x<2 \\ 4 & x \geq 2 \end{array}\right.$$

In Exercises \(53-76\), graph the piccewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant. $$f(x)=\left\\{\begin{array}{ll} x & x<0 \\ x^{2} & x \geq 0 \end{array}\right.$$

Graph the piecewise-defined function to determine whether it is a one-to-one function. If it is a one-to-one function, find its inverse. $$G(x)=\left\\{\begin{array}{ll} 0 & x<0 \\ \sqrt{x} & x \geq 0 \end{array}\right.$$

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=4 x^{2}-9, \quad g(x)=\frac{\sqrt{x+9}}{2} \text { for } x \geq 0$$

Graph the function using transformations. $$y=\frac{1}{x+3}+2$$

In Exercises \(53-76\), graph the piccewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant. $$f(x)=\left\\{\begin{array}{ll} -x & x \leq 0 \\ x^{2} & x>0 \end{array}\right.$$

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\sqrt[3]{8 x-1}, \quad g(x)=\frac{x^{3}+1}{8}$$

Graph the piecewise-defined function to determine whether it is a one-to-one function. If it is a one-to-one function, find its inverse. $$G(x)=\left\\{\begin{array}{ll} \frac{1}{x} & x<0 \\ \sqrt{x} & x \geq 0 \end{array}\right.$$

In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\frac{1}{x-1}, \quad g(x)=\frac{x+1}{x} \text { for } x \neq 0, x \neq 1$$