Question

the graph of $y = f(x)$ is shown below. find all values of $x$ where $f(x) = -5$.

Explanation:

Step1: Understand the problem

We need to find the \( x \)-value(s) where \( f(x) = -5 \) from the graph of \( y = f(x) \). This means we look for the point(s) on the graph where the \( y \)-coordinate is \( -5 \), and then find the corresponding \( x \)-coordinate(s).

Step2: Analyze the graph

The graph is a straight line. We can see that when \( y = -5 \), we need to find the \( x \)-value. Let's first find the equation of the line to verify or directly read from the graph. The line passes through some points. Let's find two points: when \( x = 0 \), \( y = -4 \) (the \( y \)-intercept). Let's take another point. Let's see the slope. From \( x = 0 \), \( y = -4 \) and let's see when \( x = 1 \), \( y \) is \( -4.5 \)? Wait, maybe better to use the fact that we need \( y = -5 \). Let's solve for \( x \) when \( y = -5 \).

First, find the slope \( m \). Let's take two points: when \( x = -10 \), \( y = -1 \) (from the left end) and \( x = 0 \), \( y = -4 \). The slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-1)}{0 - (-10)} = \frac{-3}{10} = -0.3 \). So the equation of the line is \( y = -0.3x - 4 \) (since the \( y \)-intercept is \( -4 \)).

Now set \( y = -5 \):

\( -5 = -0.3x - 4 \)

Step3: Solve for \( x \)

Add 4 to both sides:

\( -5 + 4 = -0.3x \)

\( -1 = -0.3x \)

Divide both sides by \( -0.3 \):

\( x = \frac{-1}{-0.3} = \frac{10}{3} \approx 3.333 \)? Wait, that doesn't seem right. Wait, maybe I misread the graph. Wait, looking at the graph again, the line is from the left ( \( x = -10 \), \( y = -1 \)) going down to the right. Wait, maybe my initial point is wrong. Wait, when \( x = 0 \), \( y = -4 \). Let's take another point: when \( x = 10 \), what's \( y \)? Wait, the line is going towards the right, decreasing. Wait, maybe the slope is \( -\frac{1}{2} \)? Wait, no, let's check the grid. Each grid square is 1 unit. So from \( x = 0 \), \( y = -4 \), and when \( x = 2 \), \( y = -5 \)? Wait, no, let's look at the graph again. Wait, the user's graph: let's see, the \( y \)-axis has -1, -2, -3, -4, -5, etc. The line is a straight line. Let's find when \( y = -5 \). Let's see the difference between \( y = -4 \) (at \( x = 0 \)) and \( y = -5 \) is a decrease of 1 in \( y \). Since the slope: let's see, from \( x = 0 \), \( y = -4 \), and if we move 2 units to the right ( \( x = 2 \) ), does \( y \) decrease by 1? Wait, maybe the slope is \( -\frac{1}{2} \). Wait, no, let's do it properly.

Wait, maybe the line passes through \( (0, -4) \) and \( (2, -5) \). So the slope \( m = \frac{-5 - (-4)}{2 - 0} = \frac{-1}{2} = -0.5 \). So the equation is \( y = -0.5x - 4 \). Now set \( y = -5 \):

\( -5 = -0.5x - 4 \)

Add 4 to both sides:

\( -1 = -0.5x \)

Multiply both sides by -2:

\( x = 2 \)

Wait, but let's check the graph. If \( x = 2 \), \( y = -5 \)? Let's see, the graph: when \( x = 0 \), \( y = -4 \); when \( x = 2 \), moving 2 units right, slope is -0.5, so \( y \) decreases by 1, so \( y = -5 \). Yes, that makes sense. So the \( x \)-value when \( y = -5 \) is \( x = 2 \).

Wait, but maybe I made a mistake. Let's look at the graph again. The line is going from the left ( \( x = -10 \), \( y = -1 \)) to the right, decreasing. Wait, when \( x = -10 \), \( y = -1 \); when \( x = 0 \), \( y = -4 \). So the change in \( x \) is 10, change in \( y \) is -3. So slope \( m = \frac{-3}{10} = -0.3 \). Then equation is \( y = -0.3x - 4 \). Set \( y = -5 \):

\( -5 = -0.3x - 4 \)

\( -1 = -0.3x \)

\( x = \frac{1}{0.3} \approx 3.333 \). But that doesn't seem to match. Wait, maybe the graph is different. Wait, the user's graph: the line starts at \( x = -10 \), \( y = -1 \) (the left end) and goes to the right, passing through \( (0, -4) \), and then to the right. Wait, maybe the correct \( x \) when \( y = -5 \) is \( x = 2 \). Wait, maybe my initial point is wrong. Let's look at the graph again. The \( y \)-axis: at \( x = 0 \), \( y = -4 \). Then, each grid square is 1 unit. So to get \( y = -5 \), we need to go down 1 unit from \( y = -4 \). Since the line is decreasing, we move to the right. How much? Let's see the slope. From \( x = 0 \), \( y = -4 \), and if we move 2 units to the right ( \( x = 2 \) ), \( y \) becomes \( -5 \). So \( x = 2 \).

Wait, but let's check the graph. If we look at the graph, when \( y = -5 \), the \( x \)-value is 2. So the answer is \( x = 2 \).

Answer:

\( x = 2 \)