How to Find an Inverse Function

An inverse function reverses what the original function does. If $f$ takes an input and produces an output, then the inverse $f^{-1}$ takes that output and returns the original input, so $f(a)=b$ means exactly $f^{-1}(b)=a$.

Because of this undoing, composing a function with its inverse gives back the input unchanged: $ff^{-1}(x)=x$ and $f^{-1}f(x)=x$. This method answers questions such as "find $f^{-1}(x)$", "evaluate $f^{-1}(7)$", or "state the value of $x$ for which $f^{-1}$ is undefined".

It is a core Form 4 Functions skill. A function only has an inverse if it is one-to-one, each output comes from exactly one input, which is why the topic is closely tied to domain and range.

Thinking of $f^{-1}$ as a reverse machine keeps the algebra honest. Whatever $f$ does last, $f^{-1}$ must undo first.

This picture also explains why the graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y=x$.

Reach for this method whenever a question uses the symbol $f^{-1}$ and asks you to find it as an expression, to evaluate it at a number, or to state where it is defined. Typical wording is "find $f^{-1}(x)$" or "hence find $f^{-1}(5)$".

You also use it when a question asks you to show two functions are inverses of each other, which you confirm by checking that their composite is $x$. Another cue is a graph question asking for a reflection in $y=x$.

Because Functions opens the syllabus, inverse-function work appears in both Paper 1 and Paper 2, often linked to composite functions. If you see $f^{-1}$ anywhere, this is the tool, but first make sure the function is one-to-one so the inverse actually exists.

Follow this order for a clean inverse every time:

1. 1

   Write $y=f(x)$

   Replace the function name with $y$, so a rule like $f(x)=3x+1$ becomes $y=3x+1$.

2. 2

   Make $x$ the subject

   Rearrange the equation to get $x$ alone on one side, undoing each operation in reverse order.

3. 3

   Swap $y$ for $x$

   Replace every $y$ with $x$ and write the result as $f^{-1}(x)$. This gives the inverse in standard form.

4. 4

   State any restriction

   If the inverse involves a fraction or a square root, note the values of $x$ it excludes; the domain of $f^{-1}$ equals the range of $f$.

5. 5

   Check with a composite

   Confirm your answer by showing $ff^{-1}(x)=x$. If it simplifies to $x$, the inverse is correct.

Try this yourself first, then check each line against the solution.

Notice the two independent checks in the note. Showing $ff^{-1}(x)=x$ confirms the general rule, while $f(3)=10$ confirms the single value.

When both agree, you can move on without doubt.

• Confusing $f^{-1}(x)$ with $\frac{1}{f(x)}$. The inverse is not a reciprocal; the $-1$ is a notation for "reverse", not a power.
• Undoing operations in the wrong order. Reverse the order: if $f$ multiplies then adds, undo by subtracting then dividing.
• Forgetting to swap $y$ and $x$ at the end, leaving the answer in terms of $y$.
• Missing the domain restriction when the inverse has a fraction or a root, so the excluded value is not stated.
• Trying to invert a function that is not one-to-one; without a suitable domain restriction the inverse does not exist.

The single step that costs marks is the rearrangement, the moment you make $x$ the subject and one operation gets undone out of order. In a one-to-one lesson our teacher watches that exact line, sees whether you divided before subtracting or forgot to swap the variables, and has you redo just that step until it is automatic.

Because our teachers are experienced, you get someone who links the algebra back to the "reverse machine" idea so it makes sense, not just sticks. Lessons are online and taught in English, while SPM papers are set in both Malay and English.

We also build the habit of the composite check, $ff^{-1}(x)=x$, so a wrong inverse is caught before it reaches the answer line.