Section 2.9: Inverse Functions

  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. Does the following function $f$ have an inverse? If yes, find the values of (a) $f(-1)$,  (b) $f^{-1}(-2)$,   (c) $f^{-1}(1)$.   solution

    3. figure

  3. Find the values of (a) $f^{-1}(1)$,  (b) $f^{-1}(3)$,   (c) $f^{-1}(4)$.   solution

    4. figure


    Additional Questions (Optional)

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

  5. 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

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

  7. 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

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

    7. The function \( f(x)=\dfrac{4x+1}{2x-3} \) is one-to-one. Find its inverse, state the domain and range of \( f \). Also 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. Find the inverse of the following one-to-one function. \(f(x)=\dfrac{4x+1}{2x-3} \).  solution