Piecewise Function Math Example 4

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Example 4

hard
Find the value of cc so that f(x)={cx+1x≤3x2−2x>3f(x) = \begin{cases} cx + 1 & x \leq 3 \\ x^2 - 2 & x > 3 \end{cases} is continuous at x=3x = 3.

Solution

  1. 1
    For continuity, left limit must equal right limit at x=3x=3. Right limit: lim⁡x→3+(x2−2)=9−2=7\lim_{x\to3^+}(x^2-2) = 9-2=7.
  2. 2
    Left limit: lim⁡x→3−(cx+1)=3c+1\lim_{x\to3^-}(cx+1) = 3c+1. Set equal: 3c+1=7⇒3c=6⇒c=23c+1=7 \Rightarrow 3c=6 \Rightarrow c=2.

Answer

c=2c = 2
To ensure continuity across a piecewise boundary, equate the one-sided limits. Here the right-hand piece fixes the target value 77, and we solve for cc so the left-hand piece also approaches 77 at x=3x=3.

About Piecewise Function

A piecewise function is defined by different formulas on different non-overlapping intervals of its domain, with the applicable formula determined by the input value.

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More Piecewise Function Examples

Example 1 easy

Evaluate [formula] at [formula], [formula], and [formula].

Example 2 medium

Determine whether [formula] is continuous at [formula].

Example 3 easy

Given [formula], evaluate [formula], [formula], and [formula].

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