How Do You Find the Inverse Function Algebraically?

To find an inverse function algebraically, write the function as $y=f(x)$, interchange $x$ and $y$, then rearrange to make $y$ the subject. Finally, write the result as $f^{-1}(x)$ and check that the original function is one-to-one.

This is part of IB Mathematics: Analysis and Approaches, SL 2.5: Composite functions, identity, finding inverse, and applies to both Paper 1 and Paper 2.

The Algebraic Method

An inverse function reverses the effect of the original function. Algebraically, swapping $x$ and $y$ reverses the roles of the input and output.

Use these steps:

1. Write $f(x)$ as $y$.
2. Interchange $x$ and $y$.
3. Rearrange to make $y$ the subject.
4. Replace $y$ with .

For example, let

$f(x)=\frac{3x-5}{2}.$

Write $y=\frac{3x-5}{2}$, then interchange the variables:

$x=\frac{3y-5}{2}.$

Now rearrange:

$2x=3y-5,$

$3y=2x+5,$

$y=\frac{2x+5}{3}.$

Therefore,

$f^{-1}(x)=\frac{2x+5}{3}.$

Verify using composition:

$f\left(f^{-1}(x)\right)=x$

and, where both compositions are defined,

$f^{-1}(f(x))=x.$

For a linear function $f(x)=ax+b$, where $a\neq 0$, the inverse is $f^{-1}(x)=\frac{}{}$. This also shows why is impossible: a constant function sends every input to the same output, so it cannot be one-to-one and has no inverse function.

The domain of $f^{-1}$ equals the range of $f$, while its range equals the domain of $f$. Graphically, the inverse is the reflection of the original graph in the line $y=x$, explaining why domain and range exchange roles.

For example, $f(x)=x^2$ is not one-to-one on all real numbers because $f(2)=f(-2)=4$. If its domain is restricted to , then gives , so for . This restriction selects one branch of the parabola, making each output correspond to exactly one input. Always include the restriction in the final answer.

A common misconception is that $f^{-1}(x)=\frac{1}{f(x)}$. This is false: $$ is the inverse function, not the reciprocal.

Exam Technique

For the IB command term find, show the variable swap and rearrangement rather than only the final expression. Check whether the function is one-to-one, state necessary restrictions, and use composition to verify your result.