Question

the graph of $y = f(x)$ is shown below. find all values of $x$ where $f(x) = 6$.

Explanation:

Step1: Understand the problem

We need to find the \( x \)-values where \( f(x) = 6 \). This means we look for the points on the graph of \( y = f(x) \) where the \( y \)-coordinate is 6, and then find the corresponding \( x \)-coordinates.

Step2: Analyze the graph

The graph is a straight line (linear function). We can see that the line has a slope and a \( y \)-intercept. Let's first find the equation of the line to verify, but alternatively, we can look at the grid.

Looking at the \( y \)-axis, \( y = 6 \) is a horizontal line. We need to find where the graph of \( y = f(x) \) intersects \( y = 6 \).

From the graph, we can see that when \( y = 6 \), we need to find the \( x \)-value. Let's find the equation of the line first. The line passes through \( (0, 2) \) (the \( y \)-intercept) and \( (3, 0) \) (the \( x \)-intercept). The slope \( m \) is \( \frac{0 - 2}{3 - 0} = -\frac{2}{3} \). So the equation is \( y = -\frac{2}{3}x + 2 \).

Now, set \( y = 6 \):

$$ 6 = -\frac{2}{3}x + 2 $$

Subtract 2 from both sides:

$$ 4 = -\frac{2}{3}x $$

Multiply both sides by \( -\frac{3}{2} \):

$$ x = 4 \times (-\frac{3}{2}) = -6 $$

Alternatively, by looking at the graph, when \( y = 6 \), we can see that the \( x \)-value is \( -6 \) (since moving up from \( y = 2 \) (at \( x = 0 \)) to \( y = 6 \) is 4 units up. Since the slope is \( -\frac{2}{3} \), the change in \( x \) is \( \frac{\Delta y}{m} = \frac{4}{-\frac{2}{3}} = -6 \), so \( x = 0 - 6 = -6 \).

Answer:

\( x = -6 \)