Question

which input value produces the same output value for the two functions on the graph?
$f(x)=-\frac{2}{3}x + 1$
$g(x)=\frac{1}{3}x - 2$
(graph of two linear functions f(x) and g(x) intersecting at a point, with options: $x=-1$, $x=1$, $x=3$, $x=-3$)

Explanation:

Step1: Set functions equal

To find the input \( x \) where \( f(x) = g(x) \), set \( -\frac{2}{3}x + 1=\frac{1}{3}x - 2 \).

Step2: Solve for \( x \)

Add \( \frac{2}{3}x \) to both sides: \( 1=\frac{1}{3}x+\frac{2}{3}x - 2 \).
Simplify: \( 1 = x - 2 \).
Add 2 to both sides: \( x = 3 \).

Answer:

\( x = 3 \) (corresponding to the option \( x = 3 \))