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\\)

Explanation:

Identify the given functions

We are given two linear functions:

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

$$g(x) = \frac{1}{3}x - 2$$

Understand the graphical meaning

The question asks for the input value \(x\) that produces the same output value \(y\) for both functions.
This corresponds to the \(x\)-coordinate of the intersection point of the two lines on the graph.

Locate the intersection point

Looking at the graph, the red line \(f(x)\) and the blue line \(g(x)\) intersect at a single point.
By visual inspection, this point is located at:

$$(x, y) = (3, -1)$$

Verify the intersection algebraically

We set the two function equations equal to each other to solve for \(x\):

$$-\frac{2}{3}x + 1 = \frac{1}{3}x - 2$$

Add \(\frac{2}{3}x\) to both sides:

$$1 = x - 2$$

Add \(2\) to both sides:

$$x = 3$$

Calculate the corresponding output

Substitute \(x = 3\) back into either function to find the output:

$$g(3) = \frac{1}{3}(3) - 2 = 1 - 2 = -1$$

Both functions yield the same output value of \(-1\) when the input is \(3\).

Answer:

3