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. 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