Which input value produces the same output value for the two functions on the graph?

Question

f(x) equals negative StartFraction 2 Over 3 EndFraction x plus 1. g(x) equals StartFraction 1 Over 3 EndFraction x minus 2. A coordinate grid with two lines. One line labeled f(x) passes through (negative 3, 3), (0, 1), and point (3, negative 1). The second line is labeled g(x) and passes through (negative 3, negative 3), (0, negative 2), and point (3, negative 1).

x = –3
x = –1
x = 1
x = 3

Answer 1

To find the input value that produces the same output value for the two functions \( f(x) \) and \( g(x) \), we need to set the two functions equal to each other and solve for \( x \):

1. The functions are given as:

   • \( f(x) = -\frac{2}{3}x + 1 \)
   • \( g(x) = \frac{1}{3}x - 2 \)

2. Set the two functions equal to each other: \[ -\frac{2}{3}x + 1 = \frac{1}{3}x - 2 \]

3. To eliminate the fractions, multiply through by 3: \[ -2x + 3 = x - 6 \]

4. Rearranging gives: \[ -2x - x = -6 - 3 \] \[ -3x = -9 \] \[ x = 3 \]

5. Now we will verify if \( x = 3 \) provides the same output for both functions:

   • Calculate \( f(3) \): \[ f(3) = -\frac{2}{3} \cdot 3 + 1 = -2 + 1 = -1 \]
   • Calculate \( g(3) \): \[ g(3) = \frac{1}{3} \cdot 3 - 2 = 1 - 2 = -1 \]

Since \( f(3) = g(3) = -1 \), the input value that produces the same output for both functions is:

\[ \boxed{3} \]