Periodic Functions Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
A function satisfies f(x+4)=f(x)f(x+4) = f(x) for all xx, and is defined on [0,4)[0,4) by f(x)=x2−4x+3f(x) = x^2 - 4x + 3. Find f(13.5)f(13.5).

Solution

  1. 1
    Reduce 13.513.5 modulo the period 44: 13.5=4×3+1.513.5 = 4 \times 3 + 1.5, so f(13.5)=f(1.5)f(13.5) = f(1.5).
  2. 2
    Evaluate: f(1.5)=(1.5)2−4(1.5)+3=2.25−6+3=−0.75f(1.5) = (1.5)^2 - 4(1.5) + 3 = 2.25 - 6 + 3 = -0.75.

Answer

f(13.5)=−0.75f(13.5) = -0.75
To evaluate a periodic function at any input, subtract as many full periods as needed to land in the fundamental domain, then evaluate the defining formula there.

About Periodic Functions

A function that repeats its values at regular intervals: f(x+T)=f(x)f(x + T) = f(x) for all xx, where TT is the smallest positive period.

Learn more about Periodic Functions →

More Periodic Functions Examples

Example 1 easy

Verify that [formula] is periodic with period [formula] by checking the definition [formula].

Example 2 medium

Find the period of [formula] and sketch one complete cycle.

Example 3 easy

The function [formula] has period [formula]. What is [formula] given that [formula]?

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