A reliable method
Write an intermediate variable for the inner output. To reverse an affine function, undo its operations in reverse order. Verify by composing the original and proposed inverse.
The common trap
The notation f inverse is not the reciprocal 1/f. Reversing the order of composition generally changes the result.
Make the connection
If a function repeats an output, restrict its domain before seeking an inverse function. Squaring requires a domain choice to be invertible.
Composition order changes an output
Let f(x)=2x+1 and g(x)=x². Then f(g(3))=19, whereas g(f(3))=49. The product f(3)g(3)=63 is a third operation. Write the intermediate value on a separate line so notation cannot silently change the task. To reverse f, subtract 1 and divide by 2: f⁻¹(y)=(y−1)/2. Multiplying f(y) by this expression is not how an inverse is verified; composing them is. For a square function, choose a one-to-one domain before claiming an inverse function, because x and −x share an output otherwise.
Transfer, explain and check
Find f⁻¹(11) and verify it with f. Check: 5 and f(5)=11. Then compute g(f(2))=25 and f(g(2))=9 to explain why reversing composition is not harmless.
If f(x)=3x+1 and g(x)=x², what is f(g(3))?
- Apply the inner function first: g(3)=9.
- Apply f to 9: 3(9)+1.
- The result is 28.
Compare the other answer choices
Omit the added constant in f.
Multiply f(3) and g(3) instead of composing.
Evaluate g(f(3)) instead of f(g(3)).
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus course and syllabus