LEMMA

Functions

Inverse functions

ALEKS placement

Swap x and y, then solve for y — the inverse undoes the original.

What this covers

f(f−1(x))=x
  • Check that the original function is one to one. Restrict its domain if needed so each output has only one original input.
  • Replace f(x) with y.
  • Exchange x and y.
  • Solve the new equation for y.
  • Write y = f⁻¹(x), and check that composing returns x.
  • The inverse's domain is the original range, and its range is the original domain.

Worked example

Worked example

Find the inverse of f(x) = (3x - 1)/2.

  1. x=23y−1​Swap x and y.
  2. 2x+1=3yMultiply by 2, then add 1.
  3. f−1(x)=32x+1​Divide by 3.
Another worked example

Find the inverse of f(x) = x² when the original domain is x ≥ 0.

  1. y=x2,x≥0The restricted domain keeps only one input for each output.
  2. x=y2,y≥0Swap input and output, carrying the original restriction onto y.
  3. f−1(x)=√x,x≥0Choose the nonnegative square root because the inverse must return values from the original domain.

A common mistake

A mistake Lemma catches

From

f−1(x)=3x−12​f−1(x)=32x+1​

“f⁻¹ undoes f; it is not 1 over f. Check with a number: f(3) = (9 − 1)/2 = 4, so f⁻¹(4) must be 3. (2 · 4 + 1)/3 = 3 works; 2/(3 · 4 − 1) = 2/11 does not.”

Try one

Sample problem

Find f⁻¹(x). Enter only the inverse expression.

f(x)=4x+8

Practice inverses free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

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