Question

the function $f(x)$ is shown on the graph.

graph of a function $f(x)$ on a coordinate plane with grid lines. the graph has a curve that intersects the x-axis at certain points. the question below the graph is: if $f(x) = 0$, what is $x$? with multiple - choice options: $-6$ only; $0$ only; $-6, -2, 1,$ or $3$ only; $-2, 1,$ or $3$ only

Explanation:

Step1: Understand the problem

We need to find the values of \( x \) where \( f(x) = 0 \). Geometrically, this means finding the \( x \)-intercepts of the graph of \( f(x) \), i.e., the points where the graph crosses or touches the \( x \)-axis (where \( y = 0 \)).

Step2: Analyze the graph

Looking at the graph of \( f(x) \), we observe the points where the graph intersects the \( x \)-axis. From the graph, we can see that the graph crosses the \( x \)-axis at \( x = -2 \), \( x = 1 \), and \( x = 3 \). Wait, let's check again. Wait, maybe I misread. Wait, let's look at the grid. Wait, the \( x \)-axis is the horizontal axis. Let's check the coordinates. Wait, the graph: when \( x = -2 \), does it cross the \( x \)-axis? Wait, no, maybe I made a mistake. Wait, let's re-examine. Wait, the graph: let's see the points. Wait, the graph has a root at \( x = -2 \)? Wait, no, maybe the correct \( x \)-intercepts are \( x = -2 \), \( x = 1 \), and \( x = 3 \)? Wait, no, wait the options: the options are -6 only; 0 only; -6, -2, 1, or 3 only; -2, 1, or 3 only. Wait, maybe I misread the graph. Wait, let's look again. Wait, the graph: when \( x = -2 \), does it cross the \( x \)? Wait, no, maybe the \( x \)-intercepts are at \( x = -2 \), \( x = 1 \), and \( x = 3 \)? Wait, no, wait the graph: let's see the y-intercept is at \( y = -6 \), but that's the \( y \)-intercept (when \( x = 0 \)). Wait, no, the question is \( f(x) = 0 \), so \( x \)-intercepts. So looking at the graph, the points where \( y = 0 \) (i.e., \( f(x) = 0 \)) are at \( x = -2 \), \( x = 1 \), and \( x = 3 \)? Wait, no, wait the options: the fourth option is -2, 1, or 3 only. Wait, maybe the correct \( x \)-intercepts are \( x = -2 \), \( x = 1 \), and \( x = 3 \). Wait, let's check the graph again. The graph: when \( x = -2 \), does it cross the \( x \)-axis? Let's see the grid. Each square is a unit? Let's assume each grid square is 1 unit. So the \( x \)-axis is horizontal. So the graph: let's see, at \( x = -2 \), the graph is at the bottom, then comes up, crosses the \( x \)-axis at \( x = -2 \)? Wait, no, maybe the graph crosses the \( x \)-axis at \( x = -2 \), \( x = 1 \), and \( x = 3 \). Wait, but let's check the options. The options are:

• -6 only: No, the graph doesn't cross at \( x = -6 \).
• 0 only: No, at \( x = 0 \), \( f(0) = -6 \) (from the graph, the \( y \)-intercept is -6), so \( f(0)

eq 0 \).

• -6, -2, 1, or 3 only: No, \( x = -6 \) is not an intercept.
• -2, 1, or 3 only: Yes, because the graph crosses the \( x \)-axis at \( x = -2 \), \( x = 1 \), and \( x = 3 \) (where \( f(x) = 0 \)).

Wait, maybe I made a mistake earlier. Let's confirm: the \( x \)-intercepts are the points where the graph meets the \( x \)-axis ( \( y = 0 \) ). From the graph, we can see that the graph intersects the \( x \)-axis at \( x = -2 \), \( x = 1 \), and \( x = 3 \). So the values of \( x \) where \( f(x) = 0 \) are \( x = -2 \), \( x = 1 \), and \( x = 3 \).

Answer:

-2, 1, or 3 only (the fourth option: -2, 1, or 3 only)