Step-by-Step Questions, Worked Examples, and Detailed Solutions
When a function is represented as a table of values, finding its inverse involves reversing the roles of the input and output.
If $f$ is a function with an inverse $f^{-1}$, the relationship is defined by:
\[ f(a) = b \iff a = f^{-1}(b) \]In plain English: If the original table says that $x$ maps to $y$, then the inverse maps $y$ back to $x$.
Use the table below to find the following values, if possible:
a) $f^{-1}(-4)$ | b) $f^{-1}(6)$ | c) $f^{-1}(9)$ | d) $f^{-1}(10)$ | e) $f^{-1}(-10)$
| $x$ | $f(x)$ |
|---|---|
| -4 | 9 |
| -2 | 4 |
| 0 | -5 |
| 3 | 5 |
| 6 | -4 |
| 7 | 7 |
| 8 | -10 |
a) $f^{-1}(-4)$: We look for $-4$ in the $f(x)$ column. It corresponds to $x = 6$. Thus, $f^{-1}(-4) = 6$.
b) $f^{-1}(6)$: We look for $6$ in the $f(x)$ column. It does not exist. Thus, $f^{-1}(6)$ is undefined.
c) $f^{-1}(9)$: $f(x) = 9$ when $x = -4$. Thus, $f^{-1}(9) = -4$.
d) $f^{-1}(10)$: $10$ is not in the $f(x)$ column. Thus, $f^{-1}(10)$ is undefined.
e) $f^{-1}(-10)$: $f(x) = -10$ when $x = 8$. Thus, $f^{-1}(-10) = 8$.
Use the table for function $g$ to find: $g^{-1}(0)$, $g^{-1}(-5)$, $g^{-1}(-10)$, and $g^{-1}(3)$.
| $x$ | $g(x)$ |
|---|---|
| -5 | 9 |
| -2 | 3 |
| 0 | -5 |
| 3 | 7 |
| 4 | -4 |
| 7 | 10 |
| 11 | 0 |
a) $g^{-1}(0)$: In the table, $g(11) = 0$. Therefore, $g^{-1}(0) = 11$.
b) $g^{-1}(-5)$: In the table, $g(0) = -5$. Therefore, $g^{-1}(-5) = 0$.
c) $g^{-1}(-10)$: The value $-10$ does not appear in the $g(x)$ column. Therefore, $g^{-1}(-10)$ is undefined.
d) $g^{-1}(3)$: In the table, $g(-2) = 3$. Therefore, $g^{-1}(3) = -2$.