Find an inverse and sketch its reflected graph
Reverse a one-to-one rule and use reflection in y = x to connect the function and its inverse.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What must be true before a function can have an inverse function on its domain?
What you will learn
- Find inverse rules by exchanging input and output
- Keep the domain and range consistent after inversion
- Sketch a function and its inverse as reflections in y = x
Reverse the mapping, not the formula by inspection
Write y = f(x), interchange x and y, then solve for y. The resulting rule is f⁻¹(x). Check that the original function is one-to-one on its domain before using it.
The original domain becomes the inverse range, and the original range becomes the inverse domain. These sets help detect an invalid restriction.
The graph reflects across y = x
A point (a, b) on y = f(x) becomes (b, a) on y = f⁻¹(x). Sketch the line y = x as a mirror and swap the coordinates of useful points.
An inverse graph is not generally a reflection across the x-axis. It is reflected across the diagonal y = x.
Pause and explain
What does a point (a, b) become under inverse reflection?
Worked example
Find the inverse of f(x) = 4x − 7 and state the graph reflection.
Show the worked solution
- Write y = 4x − 7.
- Exchange x and y: x = 4y − 7.
- Solve for y: y = (x + 7)/4.
Answer f⁻¹(x) = (x + 7)/4; the two graphs reflect in y = x.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the inverse of f(x) = 5x + 2, where x is real.
Give me a hint
Set y equal to the function, swap x and y, then isolate y.
Compare my reasoning
- y = 5x + 2.
- Swap: x = 5y + 2.
- Subtract 2 and divide by 5.
f⁻¹(x) = (x − 2)/5.
Look for these in your work
- Variables swapped
- Algebra solved
- Inverse notation used
For f(x) = x² with x ≥ 0, find f⁻¹ and give its domain.
Give me a hint
Use the stated restriction to choose one square-root branch.
Compare my reasoning
- Write y = x² with x ≥ 0.
- Swap variables: x = y².
- The original x ≥ 0 makes the inverse y = √x; the original range is x ≥ 0.
f⁻¹(x) = √x for x ≥ 0.
Look for these in your work
- Restriction respected
- Positive root selected
- Inverse domain stated
A calibration graph contains point (12, 4), where input voltage 12 produces output 4. What point belongs to the inverse calibration?
Give me a hint
An inverse answers the input needed to obtain a target output.
Compare my reasoning
- Swap the output and input coordinates.
- The target output 4 becomes the inverse input.
- The required original input 12 becomes the inverse output.
(4, 12): an output of 4 corresponds to an input of 12.
Look for these in your work
- Coordinates reversed
- Meaning explained
- No sign change invented
Common mistakes
- Calling 1/f(x) the inverse function
- Forgetting to exchange x and y
- Reflecting in the x-axis rather than y = x
What can you explain now?
What point on the inverse corresponds to (2, 5) on the original graph?
Compare with the explanation
(5, 2)
Inverse mappings exchange the input and output coordinates.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
State the algebraic steps for finding an inverse and the line used for graph reflection.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.