Math · College algebra · Worked example
Find the inverse of a linear function
Find the inverse of f(x) = 3x − 4 and check it.
Solve for x
Add 4 to both sides, then divide by 3.
Swap the variables
The inverse takes the output back to the input, so rename: f⁻¹(x) = (x + 4)/3.
Check with a pair of values
f(5) = 11, and the inverse takes 11 back to 5. Writing the inverse as g in Math, both values check.
See the reflection
The graphs of f and f⁻¹ are mirror images across the line y = x. They meet on that line, at (2, 2), where f(2) = 2.
Result
f⁻¹(x) = (x + 4)/3.
Your turn
Find the inverse of f(x) = (x − 1)/5.
Show the answer and explanation
f⁻¹(x) = 5x + 1.
Solve y = (x − 1)/5 for x: multiply by 5, then add 1, so x = 5y + 1. Swap the variables: f⁻¹(x) = 5x + 1. Check: f(11) = 2 and 5(2) + 1 = 11.
Keep exploring
In Graph, the point (0, −4) on f and the point (−4, 0) on f⁻¹ swap coordinates, as every pair does across the line y = x.
Return to the concept →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
See f, f⁻¹ and y = x in Graph Check the values in Math Open worked example on a board Composition in Math ReferenceYour existing work stays on this device. Examples open as editable copies.