Question

let $f(x) = -|x - 2| + 2$ and $g(x) = \frac{1}{2}x - 2$. graph the functions in the same coordinate plane. what are the solutions to $f(x) = g(x)$? enter your answers in the boxes from smallest to largest.

Explanation:

Step1: Set functions equal

$$-|x-2| + 2 = \frac{1}{2}x - 2$$

Step2: Isolate absolute value

$$-|x-2| = \frac{1}{2}x - 4$$
$$|x-2| = -\frac{1}{2}x + 4$$

Step3: Split into two cases

Case 1: $x-2 = -\frac{1}{2}x + 4$
Case 2: $x-2 = -(-\frac{1}{2}x + 4) = \frac{1}{2}x - 4$

Step4: Solve Case 1

$$x + \frac{1}{2}x = 4 + 2$$
$$\frac{3}{2}x = 6$$
$$x = 6 \times \frac{2}{3} = 4$$

Step5: Solve Case 2

$$x - \frac{1}{2}x = -4 + 2$$
$$\frac{1}{2}x = -2$$
$$x = -4$$

Step6: Verify solutions

For $x=4$: $f(4)=-|4-2|+2=0$, $g(4)=\frac{1}{2}(4)-2=0$ (valid)
For $x=-4$: $f(-4)=-|-4-2|+2=-4$, $g(-4)=\frac{1}{2}(-4)-2=-4$ (valid)

Answer:

$-4$, $4$