Inverse Functions Flashcards

All 7 cards in this deck

What is the inverse function $f^{-1}(x)$?

The function that reverses $f$: it takes an output of $f$ back to its input.

What are the steps to find $f^{-1}(x)$?

e.g. $f(x)=2x-1$

1. Write $y=f(x)$: $y=2x-1$
2. Rearrange to make $x$ the subject: $x=\frac{y+1}{2}$
3. Replace $y$ with $x$: $f^{-1}(x)=\frac{x+1}{2}$

What are two ways to find $f^{-1}(k)$ for a given number $k$?

e.g. $f^{-1}(7)$ where $f(x)=2x+1$

Find $f^{-1}(x)$ and substitute $k$, or solve $f(x)=k$.
Here $2x+1=7$ gives $f^{-1}(7)=3$.

True or false? $f^{-1}(x)$ means $\frac{1}{f(x)}$.

False. $f^{-1}$ is the inverse function, not the reciprocal.

True or false? $ff^{-1}(x)=x$ for all values of $x$ in the domain.

True. Applying a function then its inverse returns the original input.

How do you find $h^{-1}(x)$ when $h(x)=fg(x)$?

e.g. $f(x)=x^3$, $g(x)=2x+3$

Form the composite first, then invert it.
$h(x)=(2x+3)^3$, so $h^{-1}(x)=\frac{\sqrt[3]{x}-3}{2}$

What are the steps to find an unknown constant from a condition such as $f^{-1}(13)=g(2)$?

e.g. $f(x)=3x+1$, $g(x)=ax$

1. Work out the known side: $f^{-1}(13)=\frac{13-1}{3}=4$
2. Write the other side in terms of the constant: $g(2)=2a$
3. Equate and solve: $2a=4$, so $a=2$