Inverse Functions
Have you ever used a machine that does one job, and then wished you could run it backward? 🔄 That is the basic idea behind inverse functions. In this lesson, students, you will learn what inverse functions are, how to find them, how to check whether they exist, and how they connect to the rest of the Functions topic in IB Mathematics: Analysis and Approaches HL.
What an inverse function means
A function is a rule that takes an input and gives one output. If a function is written as $f(x)$, then the inverse function is written as $f^{-1}(x)$. The inverse function “undoes” the original function.
If $f(a)=b$, then the inverse sends $b$ back to $a$, so $f^{-1}(b)=a$.
This is why inverse functions are often described as reverse operations. For example, if a calculator turns $5$ into $13$ using the rule $f(x)=2x+3$, then the inverse should turn $13$ back into $5$. To find it, solve $y=2x+3$ for $x$:
$$y=2x+3$$
$$x=\frac{y-3}{2}$$
So the inverse is
$$f^{-1}(x)=\frac{x-3}{2}$$
This works because the two functions reverse each other.
Key terminology
- The domain of a function is the set of allowed inputs.
- The range is the set of possible outputs.
- For an inverse function, the domain and range swap roles.
- A function must be one-to-one to have an inverse that is also a function.
A one-to-one function is one where different inputs always give different outputs. If two different inputs produce the same output, then the inverse would not be a function because one output would map back to two different inputs.
Why not every function has an inverse function
Not every function has an inverse function. The problem happens when a function is not one-to-one. For example, consider
$$f(x)=x^2$$
If the domain is all real numbers, then $f(2)=4$ and $f(-2)=4$. Since both $2$ and $-2$ give the same output, the function is not one-to-one.
If you try to reverse it, you would get
$$y=x^2$$
$$x=\pm\sqrt{y}$$
That gives two answers, not one, so the inverse is not a function unless we restrict the domain.
For example, if we restrict the domain to $x\ge 0$, then the function becomes one-to-one. Its inverse is
$$f^{-1}(x)=\sqrt{x}$$
This kind of domain restriction is very important in IB Mathematics because many functions, especially quadratic, trigonometric, and power functions, only have inverses after the domain is limited.
A useful visual test is the horizontal line test. If any horizontal line crosses the graph more than once, then the function is not one-to-one. If every horizontal line crosses at most once, the function has an inverse function.
How to find an inverse function
Finding an inverse usually follows a clear process. students, this is a method you should practice carefully.
Step-by-step method
- Write the function as $y=f(x)$.
- Swap $x$ and $y$.
- Solve for $y$.
- Write the result as $f^{-1}(x)$.
Example 1: Find the inverse of
$$f(x)=3x-7$$
Start with
$$y=3x-7$$
Swap $x$ and $y$:
$$x=3y-7$$
Solve for $y$:
$$x+7=3y$$
$$y=\frac{x+7}{3}$$
So
$$f^{-1}(x)=\frac{x+7}{3}$$
Check the result by composing the functions.
$$f\bigl(f^{-1}(x)\bigr)=3\left(\frac{x+7}{3}\right)-7=x$$
That confirms the inverse is correct.
Example 2: Find the inverse of
$$g(x)=\frac{2x-1}{5}$$
Let
$$y=\frac{2x-1}{5}$$
Swap $x$ and $y$:
$$x=\frac{2y-1}{5}$$
Multiply by $5$:
$$5x=2y-1$$
Add $1$:
$$5x+1=2y$$
Divide by $2$:
$$y=\frac{5x+1}{2}$$
So
$$g^{-1}(x)=\frac{5x+1}{2}$$
These linear examples are often straightforward, but the same method works for many other types of functions too.
Inverses of polynomial, rational, exponential, and logarithmic functions
Inverse functions appear across the whole Functions topic. IB Mathematics: Analysis and Approaches HL expects you to connect them with different families of functions.
Polynomial functions
A simple polynomial such as
$$f(x)=x^3$$
has an inverse function on all real numbers because it is one-to-one. To find the inverse:
$$y=x^3$$
$$x=y^3$$
$$y=\sqrt[3]{x}$$
So
$$f^{-1}(x)=\sqrt[3]{x}$$
But a quadratic like
$$f(x)=x^2+1$$
is not one-to-one on all real numbers. If restricted to $x\ge 0$, then its inverse is
$$f^{-1}(x)=\sqrt{x-1}$$
with domain $x\ge 1$.
Rational functions
For a rational function, the inverse may also be rational. For example,
$$f(x)=\frac{1}{x}$$
is its own inverse, because
$$f\bigl(f(x)\bigr)=\frac{1}{1/x}=x$$
for $x\ne 0$.
Another example is
$$f(x)=\frac{ax+b}{cx+d}$$
where the inverse can often be found by algebraic rearrangement, as long as the function is one-to-one on its domain.
Exponential and logarithmic functions
Exponential and logarithmic functions are inverse pairs. This is one of the most important inverse relationships in mathematics.
If
$$f(x)=a^x$$
for $a>0$ and $a\ne 1$, then the inverse is the logarithmic function
$$f^{-1}(x)=\log_a x$$
This means:
$$\log_a(a^x)=x$$
$$a^{\log_a x}=x$$
These identities are essential in solving equations.
Example: If
$$2^x=7$$
then the solution can be written using a logarithm:
$$x=\log_2 7$$
In real-world situations, exponential and logarithmic inverses help model growth and decay. For example, if a population grows according to an exponential model, the inverse logarithmic relationship can help find the time needed to reach a certain size.
Graphs of inverse functions and reflections
The graph of $f^{-1}(x)$ is the reflection of the graph of $f(x)$ in the line
$$y=x$$
This is a powerful visual idea.
Why does this happen? Because every point $(a,b)$ on the graph of $f$ becomes $(b,a)$ on the graph of $f^{-1}$. Swapping coordinates is exactly what reflection in $y=x$ does.
Example: If $(2,5)$ is on the graph of $f$, then $(5,2)$ is on the graph of $f^{-1}$.
This reflection idea also helps you check whether your inverse is reasonable. If the graph of a function and the graph of its inverse do not look like reflections in $y=x$, something may be wrong.
Important graph features also swap:
- domain of $f$ becomes range of $f^{-1}$
- range of $f$ becomes domain of $f^{-1}$
- intercepts switch in a predictable way
If $f(0)=4$, then $f$ passes through $(0,4)$, and $f^{-1}$ passes through $(4,0)$.
Composite functions and inverse checks
Inverse functions connect strongly to composite functions. The composition of a function and its inverse gives the identity function.
$$f\bigl(f^{-1}(x)\bigr)=x$$
$$f^{-1}\bigl(f(x)\bigr)=x$$
This works whenever the input is in the correct domain.
This is not just a formal rule. It is a practical check. If you find an inverse, you can verify it by composing.
Example:
Let
$$f(x)=4x+1$$
and
$$f^{-1}(x)=\frac{x-1}{4}$$
Then
$$f\left(f^{-1}(x)\right)=4\left(\frac{x-1}{4}\right)+1=x$$
and
$$f^{-1}(f(x))=\frac{(4x+1)-1}{4}=x$$
This identity is one of the clearest ways to show a function and its inverse are correct.
Conclusion
Inverse functions are about reversing a rule. students, the central ideas to remember are that an inverse swaps inputs and outputs, only one-to-one functions have inverse functions, and the graph of a function and its inverse reflect across $y=x$. In IB Mathematics: Analysis and Approaches HL, inverse functions connect directly to algebraic solving, graphing, domain and range, and the function families you study throughout the course. They are especially important for exponential and logarithmic relationships, where inverse thinking helps solve real-world problems and equations efficiently. 🔍
Study Notes
- An inverse function reverses the action of a function.
- If $f(a)=b$, then $f^{-1}(b)=a$.
- A function must be one-to-one to have an inverse function.
- Use the horizontal line test to check whether a graph is one-to-one.
- To find an inverse, write $y=f(x)$, swap $x$ and $y$, then solve for $y$.
- The graph of $f^{-1}(x)$ is a reflection of the graph of $f(x)$ in the line $y=x$.
- Domain and range swap between a function and its inverse.
- Polynomial examples may need a restricted domain, such as $x\ge 0$ for $f(x)=x^2$.
- Exponential and logarithmic functions are inverse pairs.
- Checking $f\bigl(f^{-1}(x)\bigr)=x$ and $f^{-1}\bigl(f(x)\bigr)=x$ confirms an inverse.
- Inverse functions are used in equations, modeling, and interpreting function behavior across IB Mathematics: Analysis and Approaches HL.