The Inverse of a Function

1. Assume that the function \(f\) is one-to-one with domain: \((-\infty, \infty)\).  (a) If \( f(4)=-1\), find \(f^{-1}(-1)\),   and (b) Find \( (f \circ f^{-1})(-10) \).  solution

2. For  \( f(x)= x^3-2 \), find each  of the following. (a) \( f(2) \)     (b) \(f^{-1}(62) \)      (c) \( (f \circ f^{-1})(235)\).  solution

3. Consider the functions \( f(x) = 2-3x \) and \(g(x)=\frac{2-x}{3} \).  (a) Find \( f(g(x)) \),   (b) Find \( g(f(x)) \), (c) Determine whether the functions \(f\) and \(g\) are inverses of each other.  solution

4. Show by finding compositions that the inverse of \( f(x)= 3x-7\) is \(f^{-1}(x)=\frac{x+7}{3}\).  solution

5. The function \(f(x)=\frac{2}{x-5}\) is one-to-one.  (a) Find the inverse of \(f\).  (b) State the domain and range of \(f\).   (c) State the domain and range of \(f^{-1}\).  (d) Graph \(f,  f^{-1}\) and \(y=x\) on the same set of axes.  solution

6. Find the inverse of the function \(f(x) = \sqrt[3]{5x-2}\) and its domain and range.  solution

7.  (a) Determine if the function \( f(x)=\frac{4x+1}{2x-3} \) is one-to-one. (b) If it is one-to-one, find its inverse.  (c) State the domain and range of \( f \).  (d) Find the domain and the range of \(f^{-1}\).  solution

8. Assume the function \( \displaystyle{f(x)=\frac{x-5}{x+6}}\) is one-to-one. Find its inverse \(f^{-1}(x)\), and the domain and range of the function. Also find the domain and range of \(f^{-1}(x)\).  solution

9. The graph of \(y=f(x)\) is given. On the same set of axes draw its inverse.  solution

10. The graph of \(y=f(x)\) is given. On the same set of axes draw its inverse.  solution