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Math · 20 · Advanced Math

Function Basics

Notes

Key points

  • f(x)f(x) is the output at xx — not ff times xx.
  • Evaluate f(3)f(3): replace every xx with (3)(3), brackets included.
  • Solve f(x)=7f(x)=7: find the input that gives output 7.
  • Domain = allowed inputs. Range = outputs actually produced.
  • Domain breaks: denominator =0=0, or an even root of a negative.
  • Composition f(g(x))f(g(x)): work inside out — gg first, then ff.
  • Tables count too: input in the x-column, output in the f(x)-column.

Formulas

Evaluatingf(x)=x2−3x ⇒ f(−2)=(−2)2−3(−2)=10f(x)=x^2-3x\ \Rightarrow\ f(-2)=(-2)^2-3(-2)=10
Substituting an expressionf(x)=2x+5 ⇒ f(x+1)=2(x+1)+5=2x+7f(x)=2x+5\ \Rightarrow\ f(x+1)=2(x+1)+5=2x+7
Composition(f∘g)(x)=f(g(x))(f\circ g)(x)=f\big(g(x)\big)Inner function first
Domain restrictions1x−a: x≠ax−a: x≥a\dfrac{1}{x-a}:\ x\neq a\qquad \sqrt{x-a}:\ x\geq a
Linear f from two valuesm=f(x2)−f(x1)x2−x1m=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}

Common traps

  • Writing f(−2)f(-2) with −22=−4-2^2=-4; the bracket gives (−2)2=4(-2)^2=4.
  • Mixing up f(g(2))f(g(2)) and g(f(2))g(f(2)) — order matters.
  • Answering with xx when the question asks for f(x)f(x), or the reverse.

SAT tip

Type the function into Desmos, then f(3) or f(g(2)) on a new line — it evaluates instantly.

Test yourself

10 official SAT questions on Function Basics, easiest first.

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