LEMMA

Functions

Function composition

ALEKS placement

(f ∘ g)(x) means run g first, then feed its output into f.

What this covers

(f∘g)(x)=f(g(x))
  • Work from the inside out.
  • Substitute the entire inner expression into every x of the outer rule.
  • Simplify only after the substitution is complete.
  • Order matters: f(g(x)) rarely equals g(f(x)).
  • Keep inputs that the inner function accepts and whose outputs the outer function also accepts.

Worked example

Worked example

If f(x) = 2x + 1 and g(x) = x², find f(g(3)).

  1. g(3)=32The inner function runs first. Replace its input with three before squaring.
  2. g(3)=9Square the input to find the inner function's output.
  3. f(g(3))=f(9)Replace the whole inner function value with its output, nine.
  4. f(9)=2⋅9+1Use nine as the input to the outer rule: double that input, then add one.
  5. f(9)=19Evaluate the outer rule to finish the composition.
Another worked example

For f(x) = 2x + 1 and g(x) = x², compare f(g(x)) with g(f(x)).

  1. f(g(x))=2(x2)+1For f after g, put the output x² into the input slot of f.
  2. f(g(x))=2x2+1The outer rule doubles the whole inner output, then adds one.
  3. g(f(x))=(2x+1)2For g after f, square the entire output of f, not just its first term.
  4. g(f(x))=4x2+4x+1Expanding the binomial shows why reversing the order changes the result.

A common mistake

A mistake Lemma catches

From g(3)=9

f(g(3))=7⋅9f(g(3))=2⋅9+1

“f(g(3)) means put g(3) into f, not multiply f(3) by g(3). g(3) = 9, so f(g(3)) = f(9) = 2 · 9 + 1 = 19. Multiplying gives f(3) · g(3) = 7 · 9 = 63.”

Try one

Sample problem

If f(x) = x − 2 and g(x) = 2x, find f(g(-4)).

f(x)=x−2,g(x)=2x

Practice composition 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

Before this

After this