Home › Functions and graphs

How to add piecewise functions with different breakpoints

Combine all breakpoints first

If two piecewise functions switch formulas at different places, list every breakpoint from both functions. On each resulting interval, choose the applicable formula from each function and add them. Check strict and inclusive inequalities at each breakpoint.

Problem

Let $f (x) = {5 x + 3, x^{2} + 8 x, x < 1 x \geq 1$ and $g (x) = {- 2 x + 1, x - 6, x \leq 0 x > 0$ . Find $(f + g) (x)$ .

Answer

(f + g) (x) = ⎩ ⎨ ⎧ 3 x + 4, 6 x - 3, x^{2} + 9 x - 6, x \leq 0 0 < x < 1 x \geq 1

Both boundary points use the inclusive branch stated in the original definitions.

Step-by-step solution

1. List breakpoints

The formula for $g$ changes at $0$ ; the formula for $f$ changes at $1$ . So use three intervals: $x \leq 0$ , $0 < x < 1$ , and $x \geq 1$ .

2. Add the formulas on each interval

For $x \leq 0$ , use $5 x + 3$ and $- 2 x + 1$ , giving $3 x + 4$ . For $0 < x < 1$ , use $5 x + 3$ and $x - 6$ , giving $6 x - 3$ . For $x \geq 1$ , use $x^{2} + 8 x$ and $x - 6$ , giving $x^{2} + 9 x - 6$ .

3. Check endpoints

At $x = 0$ , $g (0) = 1$ , so the first branch gives $(f + g) (0) = 4$ . At $x = 1$ , $f (1) = 9$ and $g (1) = - 5$ , so the third branch gives $(f + g) (1) = 4$ .

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.