What an inverse function means

If $f(9)=7$, then $f^{-1}$ sends $7$ back to $9$, so $f^{-1}(7)=9$. In general $f^{-1}(f(a))=a$: applying $f$ and then $f^{-1}$ brings you back.

$y=x^2$ $(x\ge 0)$ and $y=\sqrt{x}$ are inverse functions of each other. Squaring and then taking the square root returns the original number.

The graphs of $y=f(x)$ and $y=f^{-1}(x)$ are symmetric about the line $y=x$.

Examples

1. 01If f(9) = 7, what is the value of the inverse function?$\displaystyle f^{-1}(7)$AnswerHide

   Answer$\displaystyle 9$

   Steps

   1. $f$ sends 9 to 7, so $f^{-1}$ sends 7 back to 9
   2. $\displaystyle f(9)$$\displaystyle = 7$$\displaystyle f^{-1}(7)$$\displaystyle = 9$
2. 02Square, then take the root: you get back the original. What is the value?$\displaystyle \sqrt{4^2}$AnswerHide

   Answer$\displaystyle 4$
3. 03If f(4) = 2, what is the value of the inverse function?$\displaystyle f^{-1}(2)$AnswerHide

   Answer$\displaystyle 4$

   Steps

   1. $f$ sends 4 to 2, so $f^{-1}$ sends 2 back to 4
   2. $\displaystyle f(4)$$\displaystyle = 2$$\displaystyle f^{-1}(2)$$\displaystyle = 4$
4. 04If f(7) = 3, what is the value of the inverse function?$\displaystyle f^{-1}(3)$AnswerHide

   Answer$\displaystyle 7$

   Steps

   1. $f$ sends 7 to 3, so $f^{-1}$ sends 3 back to 7
   2. $\displaystyle f(7)$$\displaystyle = 3$$\displaystyle f^{-1}(3)$$\displaystyle = 7$
5. 05If f(9) = 4, what is the value of the inverse function?$\displaystyle f^{-1}(4)$AnswerHide

   Answer$\displaystyle 9$

   Steps

   1. $f$ sends 9 to 4, so $f^{-1}$ sends 4 back to 9
   2. $\displaystyle f(9)$$\displaystyle = 4$$\displaystyle f^{-1}(4)$$\displaystyle = 9$
6. 06If f(6) = 5, what is the value of the inverse function?$\displaystyle f^{-1}(5)$AnswerHide

   Answer$\displaystyle 6$

   Steps

   1. $f$ sends 6 to 5, so $f^{-1}$ sends 5 back to 6
   2. $\displaystyle f(6)$$\displaystyle = 5$$\displaystyle f^{-1}(5)$$\displaystyle = 6$

FAQ

Q1Does $f^{-1}(x)$ mean $\frac{1}{f(x)}$?AnswerHide

A

No. The $-1$ in $f^{-1}$ marks the inverse function, not a reciprocal. If $f(4)=2$, then $f^{-1}(2)=4$, not $\frac{1}{2}$.

Q2Isn't $\sqrt{(-3)^2}$ equal to $-3$?AnswerHide

A

No, $\sqrt{9}=3$. The square root gives a value of at least $0$, so $\sqrt{a^2}=a$ holds only for $a\ge 0$. In general, $\sqrt{a^2}=|a|$.