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Magazine Problem 53
In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\sqrt{x-1}, \quad g(x)=x^{2}+1 \text { for } x \geq 1$$
Problem 54
In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=2-x^{2}, \quad g(x)=\sqrt{2-x} \text { for } x \leq 2$$
Problem 54
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. $$f(x)=\left\\{\begin{array}{ll} -x & x<-1 \\ -1 & x \geq-1 \end{array}\right.$$
Problem 54
Graph the function using transformations. $$y=\sqrt{2-x}+3$$
Problem 54
The function \(f\) is one-to-one. Find its inverse, and check your answer. State the domain and range of both \(f\) and \(f^{-1}\) $$f(x)=\frac{x}{\sqrt{x+1}}$$
Problem 55
Graph the function using transformations. $$y=\sqrt[3]{x-1}+2$$
Problem 55
In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=\frac{1}{x}, \quad g(x)=\frac{1}{x} \text { for } x \neq 0$$
Problem 55
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. $$f(x)=\left\\{\begin{array}{ll} 1 & x<-1 \\ x^{2} & x \geq-1 \end{array}\right.$$
Problem 56
In Exercises \(51-60\), show that \(f(g(x))=x\) and \(g(f(x))=x\). $$f(x)=(5-x)^{1 / 3}, \quad g(x)=5-x^{3}$$
Problem 56
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. $$f(x)=\left\\{\begin{array}{ll} x^{2} & x<2 \\ 4 & x \geq 2 \end{array}\right.$$
