Inverse Functions Practice

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1. $f(x)=3x-2$. Find $f^{-1}(x)$
   Solution
   $\frac{x+2}{3}$
   
   
2. $f(x)=(x-2)^2, x\geq 2$
   1. Find $f^{-1}(x)$
      Solution
      $\sqrt{2}+2$
      
      
   2. Domain and range of $f(x)$
      Solution
      $x\geq2$ and $y\geq0$
      
      
   3. Domain and range of $f^{-1}(x)$
      Solution
      $x\geq0$ and $y\geq2$
      
      
3. The point $(2,3)$ is on $f(x)$
   1. What point must be on $f^{-1}(x)$?
      Solution
      $(3,2)$
      
      
   2. What point must be on $y=2f\left(\frac{x}{2}-1\right)$?
      Solution
      $(6,6)$
      
      
   3. What point must be on $3-3f^{-1}(2x)$?
      Solution
      $\left(\frac{3}{2},-3\right)$
      
      
4. See $f(x)$ below:
   
   Sketch $f^{-1}(x)$ on top of the graph above
   Solution
   
   
   
   
5. In general, $f(x)$ reflects $f^{-1}(x)$ along which line?
   Solution
   $y=x$
   
   
6. Simplify $y=f\left(f^{-1}(x)\right)$
   Solution
   $5$
   
   
7. $g(x)=x^2$ and $f(x)=x^{1/2}$
   $h(x)=g\left(f(x)\right)$. True or False: Since $g(x)$ and $f(x)$ “undo” each other, $h(-4)=-4$
   Solution
   False, $g$ has an undefined input
8. $f(x)=3e^{x-2}$. Find $f^{-1}(x)$
   Solution
   $\ln x-\ln3+2$
   
   
9. Challenge: $f(x)=x^2+6x+2$. Find $f^{-1}(x)$
   Solution
   Completing the square on $y$ we get the relation:
   $f^{-1}(x)=\pm\sqrt{x+7}-3$