Inverse Functions

students, imagine a machine that turns a number into another number 🔁. If you know the output and want to find the input, you are working with the inverse of that machine. In this lesson, you will learn how inverse functions reverse the action of a function, why not every function has an inverse, and how to find and interpret inverses in IB Mathematics Analysis and Approaches SL.

What is an inverse function?

A function takes an input and produces an output. If a function is written as $f(x)$, then its inverse is written as $f^{-1}(x)$. The key idea is reversal: if $f$ sends $x$ to $y$, then $f^{-1}$ sends $y$ back to $x$.

This means that if $f(a)=b$, then $f^{-1}(b)=a$. The inverse function “undoes” the original function. Think of a vending machine and a receipt system 🍫. If the machine gives you a snack after you insert a $2$-dollar coin, the inverse would be the rule that tells you what coin was needed when you know the snack-and-change result.

For a function to have an inverse function, it must be one-to-one. This means each output value comes from exactly one input value. If two different inputs give the same output, then the inverse would not know which input to return.

For example, the function $f(x)=x^2$ is not one-to-one if its domain is all real numbers, because $f(2)=4$ and $f(-2)=4$. If you only know the output $4$, you cannot tell whether the input was $2$ or $-2$. So $f(x)=x^2$ does not have an inverse function on all real numbers. But if the domain is restricted to $x\ge 0$, then it does have an inverse.

Notation and why it matters

The notation $f^{-1}(x)$ does not mean $\dfrac{1}{f(x)}$. That is a very common mistake. The expression $f^{-1}(x)$ means the inverse function of $f$, while $\dfrac{1}{f(x)}$ means the reciprocal of the value of the function.

For example, if $f(x)=3x+1$, then the inverse is $f^{-1}(x)=\dfrac{x-1}{3}$. But the reciprocal function would be $\dfrac{1}{3x+1}$, which is completely different.

This difference is important in exams and in real problem-solving. students, always read the notation carefully ✍️.

How to find an inverse function

There are two main methods used in IB Mathematics Analysis and Approaches SL.

Method 1: Swap $x$ and $y$

Start with $y=f(x)$. Then swap $x$ and $y$, and solve for $y$.

Example: find the inverse of $f(x)=3x-5$.

1. Write $y=3x-5$.
2. Swap $x$ and $y$: $x=3y-5$.
3. Solve for $y$: $x+5=3y$, so $y=\dfrac{x+5}{3}$.
4. Therefore, $f^{-1}(x)=\dfrac{x+5}{3}$.

Check the result:

$$f\bigl(f^{-1}(x)\bigr)=3\left(\frac{x+5}{3}\right)-5=x+5-5=x$$

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

This confirms the inverse is correct.

Method 2: Use the idea of reversing operations

Some functions are easy to invert by undoing steps in reverse order.

Example: $f(x)=\dfrac{x-4}{2}$.

To reverse it, first multiply by $2$, then add $4$. So the inverse is $f^{-1}(x)=2x+4$.

This method is especially helpful with linear, exponential, and logarithmic functions.

Domain, range, and one-to-one behavior

An inverse function swaps the roles of domain and range.

If $f$ has domain $D$ and range $R$, then $f^{-1}$ has domain $R$ and range $D$.

That makes sense because inputs and outputs switch places. If a function only produces certain outputs, then its inverse can only accept those values as inputs.

One-to-one behavior is often checked using the horizontal line test. If every horizontal line intersects the graph of a function at most once, the function is one-to-one and has an inverse.

Example: the graph of $y=x^3$ passes the horizontal line test, so it has an inverse. Its inverse is $y=\sqrt[3]{x}$.

Example: the graph of $y=x^2$ fails the horizontal line test for all real $x$, because many horizontal lines hit it twice. To get an inverse, we restrict the domain to $x\ge 0$, giving the inverse $y=\sqrt{x}$.

This idea links inverse functions to transformations and graphing. Restricting a domain is a type of adjustment that makes a function usable for inversion.

Graphs of inverse functions

The graph of $y=f^{-1}(x)$ is the reflection of the graph of $y=f(x)$ in the line $y=x$.

Why? Because the points swap coordinates. If $(a,b)$ lies on the graph of $f$, then $(b,a)$ lies on the graph of $f^{-1}$. Reflection in $y=x$ does exactly that.

Example: if $f(2)=7$, then the point $(2,7)$ is on the graph of $f$. On the inverse graph, the point $(7,2)$ appears.

This graph idea is useful for checking answers. If your inverse is correct, the graph should look like the original reflected across $y=x$.

In exams, sketching both graphs can help you understand domain restrictions and whether the inverse exists over the required interval.

Inverses of common function types

Linear functions

A linear function $f(x)=ax+b$ with $a\ne 0$ always has an inverse.

Its inverse is found by solving:

$$y=ax+b$$

$$x=ay+b$$

$$y=\frac{x-b}{a}$$

So $f^{-1}(x)=\dfrac{x-b}{a}$.

Example: if $f(x)=4x-9$, then $f^{-1}(x)=\dfrac{x+9}{4}$.

Quadratic functions

A quadratic function usually does not have an inverse on all real numbers because it is not one-to-one. However, after restricting its domain, it can have an inverse.

Example: $f(x)=x^2$ for $x\ge 0$ has inverse $f^{-1}(x)=\sqrt{x}$.

If the domain were $x\le 0$, the inverse would be $f^{-1}(x)=-\sqrt{x}$.

Exponential and logarithmic functions

Exponential and logarithmic functions are inverse pairs.

If $f(x)=a^x$ with $a>0$ and $a\ne 1$, then its inverse is $f^{-1}(x)=\log_a x$.

Example: if $f(x)=2^x$, then $f^{-1}(x)=\log_2 x$.

This pair is important in modelling growth and decay, such as population growth, compound interest, and half-life 📈.

Example: if a bacterial population grows according to $P(t)=100\cdot 2^t$, then to find the time when the population reaches $800$, you solve the inverse process:

$$800=100\cdot 2^t$$

$$8=2^t$$

$$t=3$$

So the inverse idea helps answer “when?” questions from “how many?” data.

Composite functions and inverse checks

A powerful way to test whether two functions are inverses is through composition.

If $f$ and $g$ are inverses, then:

$$f\bigl(g(x)\bigr)=x$$

$$g\bigl(f(x)\bigr)=x$$

This is the mathematical version of “doing and undoing” the same action.

Example: let $f(x)=3x-5$ and $g(x)=\dfrac{x+5}{3}$.

Then:

$$f\bigl(g(x)\bigr)=3\left(\frac{x+5}{3}\right)-5=x$$

$$g\bigl(f(x)\bigr)=\frac{(3x-5)+5}{3}=x$$

So $g=f^{-1}$.

This connects inverses to composite functions in the Functions topic. Composition shows how functions combine, and inverses show how one function can completely reverse another.

Solving problems with inverse functions

Inverse functions are useful whenever you need to reverse a process.

Example: suppose $f(x)=\dfrac{5x-1}{2}$. Find the value of $x$ such that $f(x)=9$.

You can use the inverse:

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

Then:

$$x=f^{-1}(9)=\frac{2(9)+1}{5}=\frac{19}{5}$$

This is often faster than solving the equation from scratch.

Another application is checking whether a function is invertible on a given interval. If a graph is increasing everywhere on that interval, it is one-to-one and usually invertible there. If it changes direction, it may fail the test unless the domain is restricted.

Conclusion

Inverse functions are an important part of the Functions topic because they show how one rule can be undone by another. students, the key ideas to remember are one-to-one behavior, swapping inputs and outputs, domain and range reversal, and graph reflection in $y=x$ ✅. In IB Mathematics Analysis and Approaches SL, inverse functions appear in algebraic solving, graph interpretation, and modelling with linear, quadratic, exponential, and logarithmic functions. Understanding inverses helps you move confidently between equations, graphs, and real-world contexts.

Study Notes

• An inverse function reverses the action of a function.
• If $f(a)=b$, then $f^{-1}(b)=a$.
• The notation $f^{-1}(x)$ means inverse function, not $\dfrac{1}{f(x)}$.
• A function must be one-to-one to have an inverse on its domain.
• The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y=x$.
• Domain and range swap when taking an inverse.
• To find an inverse, write $y=f(x)$, swap $x$ and $y$, and solve for $y$.
• Check inverses with compositions: $f\bigl(f^{-1}(x)\bigr)=x$ and $f^{-1}\bigl(f(x)\bigr)=x$.
• Linear functions always have inverses if the gradient is not $0$.
• Quadratic functions need domain restrictions to have inverses.
• Exponential and logarithmic functions are inverse pairs.
• Inverse functions help solve “reverse” problems in modelling and equations.