What Is an Inverse Function?
An inverse function undoes what the original function does. If a function f takes an input x and produces an output y, then the inverse function f⁻¹ takes y as input and returns the original x. In mathematical notation, if f(x) = y, then f⁻¹(y) = x.
For example, if f(x) = 2x + 3, the function doubles the input and adds 3. The inverse function must subtract 3 and then halve the result to get back to the original input. So f⁻¹(x) = (x − 3)/2.
Inverse functions are a standard topic on the Extended tier of IGCSE Maths and appear in Paper 2 and Paper 4. Questions typically ask you to find the inverse of a given function, evaluate the inverse at a specific value, or solve problems involving composite and inverse functions together.
The Step-by-Step Method
Finding an inverse function follows a consistent procedure:
- Step 1: Write y = f(x). Replace f(x) with y.
- Step 2: Swap x and y. Replace every y with x and every x with y.
- Step 3: Rearrange to make y the subject.
- Step 4: Write the result as f⁻¹(x) = …
This method works for all functions at the IGCSE level, from simple linear functions to more complex rational and quadratic functions.
Example 1: Linear Function
Find f⁻¹(x) when f(x) = 5x − 7.
Step 1: y = 5x − 7
Step 2: Swap x and y: x = 5y − 7
Step 3: Rearrange: x + 7 = 5y, so y = (x + 7)/5
Step 4: f⁻¹(x) = (x + 7)/5
To verify, check that f(f⁻¹(x)) = x. f((x + 7)/5) = 5 × (x + 7)/5 − 7 = x + 7 − 7 = x. Correct.
Example 2: Function Involving a Fraction
Find f⁻¹(x) when f(x) = (3x + 1)/(x − 2).
Step 1: y = (3x + 1)/(x − 2)
Step 2: x = (3y + 1)/(y − 2)
Step 3: Multiply both sides by (y − 2): x(y − 2) = 3y + 1. Expand: xy − 2x = 3y + 1. Collect y terms on one side: xy − 3y = 2x + 1. Factor out y: y(x − 3) = 2x + 1. Divide: y = (2x + 1)/(x − 3).
Step 4: f⁻¹(x) = (2x + 1)/(x − 3)
This type of rational function inverse is common at IGCSE and requires careful algebraic manipulation. The key is collecting all the y terms on one side and factoring y out.
Example 3: Function with a Square Root
Find f⁻¹(x) when f(x) = √(2x + 5) for x ≥ −2.5.
Step 1: y = √(2x + 5)
Step 2: x = √(2y + 5)
Step 3: Square both sides: x² = 2y + 5. Rearrange: 2y = x² − 5, so y = (x² − 5)/2.
Step 4: f⁻¹(x) = (x² − 5)/2
Note that the domain of f⁻¹ is restricted to x ≥ 0 because the range of f is y ≥ 0 (a square root always gives a non-negative result).
Domain and Range of Inverse Functions
An important concept that Cambridge tests is the relationship between the domain and range of a function and its inverse:
- The domain of f becomes the range of f⁻¹
- The range of f becomes the domain of f⁻¹
This means that if f(x) is defined for x > 0 and produces outputs y ≥ 3, then f⁻¹(x) is defined for x ≥ 3 and produces outputs greater than 0.
Understanding this relationship is crucial for answering questions about when an inverse function exists and what its domain restrictions are.
Graphical Interpretation
The graph of f⁻¹(x) is the reflection of the graph of f(x) in the line y = x. This means that if the point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f⁻¹.
Cambridge sometimes asks you to sketch the inverse function given the graph of the original function. To do this, reflect key points across the line y = x and draw the resulting curve. The line y = x acts as a mirror.
This graphical relationship also explains why the domains and ranges swap: reflecting in y = x exchanges the x-coordinates and y-coordinates of every point.
Evaluating the Inverse at Specific Values
A common exam question is to find f⁻¹(k) for a specific value k. You have two approaches:
- Method 1: Find the formula for f⁻¹(x) and substitute k.
- Method 2: Set f(x) = k and solve for x. The solution is f⁻¹(k).
Method 2 can be faster when you only need one value and finding the full inverse formula would be time-consuming.
Example: If f(x) = x³ + 2, find f⁻¹(10).
Using Method 2: x³ + 2 = 10, so x³ = 8, giving x = 2. Therefore f⁻¹(10) = 2.
Composite Functions with Inverses
Cambridge frequently tests compositions involving inverse functions, such as finding f⁻¹g(x) or gf⁻¹(x). The approach is:
- First, find the inverse function if needed
- Then, apply the composite function rule: work from the inside out
Example: If f(x) = 2x + 1 and g(x) = x², find f⁻¹g(3).
First, find g(3) = 9. Then find f⁻¹(9). Since f⁻¹(x) = (x − 1)/2, we get f⁻¹(9) = (9 − 1)/2 = 4.
Common Mistakes
Watch out for these frequent errors:
- Forgetting to swap x and y. This is the most critical step. Without it, you are just rearranging the original function.
- Algebraic errors when rearranging rational functions. Take extra care with expanding brackets and collecting terms.
- Confusing f⁻¹(x) with 1/f(x). The inverse function is not the reciprocal. f⁻¹(x) reverses the function; 1/f(x) divides 1 by the function’s output.
- Not considering domain restrictions. When the original function has a restricted domain, the inverse will have a corresponding restricted domain.
- Squaring both sides without considering the sign. When removing a square root, remember that the original square root only gives non-negative values.
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