DWG NO. 1.2 — Lesson 2 of 6

Function Notation and Evaluating Functions

Unit 1: Functions and Their Graphs · ~15 min

Reading f(x)

f(x) doesn't mean "f times x." It means "the output that f produces when you feed it x." Think of a function as a machine: whatever you put in, it processes according to its rule, and something specific comes out the other side. f(3) asks the machine what comes out when the input is 3 — you find out by substituting 3 for every x in the rule and simplifying.

Evaluating with algebraic inputs

The same substitution works even when the input isn't a number yet — it can be an expression like a+h or 2x. Substitute the whole expression everywhere the variable appears, then simplify. This skill matters beyond this lesson: it's exactly what you'll need for composing functions in 1.5 and for calculus's difference quotient later on.

Piecewise functions

A piecewise function uses different rules for different parts of the domain. To evaluate one at a specific input, first check which piece's condition the input satisfies, then apply only that piece's rule.

Guided practice

• 1.Let f(x) = −x² + 4x. Find f(3).
  Show answer

  f(3) = −(9) + 12 = 3
• 2.Let g(x) = 5 − x. Find g(2a).
  Show answer

  g(2a) = 5 − 2a. g(2a) = 5 − 2a
• 3.Let f(x) = x² + 1. Find and simplify f(x+h).
  Show answer

  f(x+h) = (x+h)² + 1 = x² + 2xh + h² + 1. f(x+h) = x² + 2xh + h² + 1
• 4.p(x) = 3x if x ≤ 1, and p(x) = x + 5 if x > 1. Find p(1) and p(5).
  Show answer

  Since 1 ≤ 1, use the first piece: p(1) = 3(1) = 3. Since 5 > 1, use the second piece: p(5) = 5 + 5 = 10. p(1) = 3  ·  p(5) = 10
• 5.Let f(x) = 2x² − 3. Find f(a) − f(1), and simplify as much as possible.
  Show answer

  f(a) = 2a² − 3, and f(1) = 2(1) − 3 = −1. So f(a) − f(1) = (2a² − 3) − (−1) = 2a² − 2. f(a) − f(1) = 2a² − 2