Question

question
find the value of $f(7)$.
$y = f(x)$
(graph of a piecewise function with a v - shaped part on the left and a line on the right, axes labeled with x from -10 to 10 and y from -10 to 10)
answer attempt 1 out of 2

Explanation:

Step1: Identify the function's behavior for \( x \geq 2 \)

The graph for \( x \geq 2 \) is a straight line. Let's find its slope and equation. We know two points: when \( x = 2 \), \( y = 0 \) (from the intersection with the x - axis), and let's take another point. Let's assume the slope \( m \). From the graph, when \( x = 8 \), let's see the y - value. Wait, alternatively, let's find the equation of the line for \( x\geq2 \). The line passes through \( (2,0) \) and let's check the slope. Let's take another point, say when \( x = 8 \), looking at the graph, the y - value seems to be 6? Wait, no, let's do it properly. The line for \( x\geq2 \): let's find two points. We see that when \( x = 2 \), \( y = 0 \); when \( x = 8 \), \( y = 6 \)? Wait, no, the graph shows that the line is increasing. Wait, actually, let's find the slope between \( (2,0) \) and \( (8,6) \)? Wait, no, maybe a better way: the line for \( x\geq2 \) has a slope. Let's check the rise over run. From \( x = 2 \) (y = 0) to \( x = 7 \), how much does y increase? Wait, first, let's find the equation of the line. Let's take two points on the line for \( x\geq2 \). Let's see, when \( x = 2 \), \( y = 0 \); when \( x = 3 \), \( y = 1 \)? Wait, no, the graph: the line starts at \( (2,0) \) and goes up. Let's calculate the slope \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take \( (2,0) \) and \( (8,6) \), then \( m=\frac{6 - 0}{8 - 2}=\frac{6}{6}=1 \). Wait, no, that can't be. Wait, maybe \( (2,0) \) and \( (7,y) \). Wait, actually, looking at the graph, the line for \( x\geq2 \) has a slope of 1? Wait, no, let's check the grid. Each square is 1 unit. So from \( x = 2 \) (y = 0) to \( x = 3 \), y goes to 1; \( x = 4 \), y = 2, etc. So the equation of the line is \( y - 0 = 1\times(x - 2) \), so \( y=x - 2 \). Let's verify: when \( x = 2 \), \( y = 0 \) (correct); when \( x = 3 \), \( y = 1 \) (correct); when \( x = 7 \), \( y=7 - 2=5 \)? Wait, no, that doesn't match. Wait, maybe the slope is 1? Wait, no, maybe I made a mistake. Wait, let's look at the graph again. The line for \( x\geq2 \): when \( x = 2 \), y = 0; when \( x = 8 \), y = 6? Wait, no, the graph shows that at \( x = 8 \), the y - coordinate is 6? Wait, no, the arrow is at (8,6)? Wait, no, the graph: the vertical axis is y, horizontal is x. Let's count the units. From \( x = 2 \) (y = 0) to \( x = 7 \), the number of units in x is \( 7 - 2 = 5 \). If the slope is 1, then y would be 5, but wait, maybe the slope is 1. Wait, let's check the equation again. Let's take the line passing through \( (2,0) \) with slope \( m = 1 \). Then the equation is \( y = x - 2 \). So when \( x = 7 \), \( y=7 - 2 = 5 \)? Wait, but let's check the graph. Alternatively, maybe the slope is 1. Wait, another way: look at the graph, when \( x = 7 \), we need to find the y - value on the line for \( x\geq2 \). The line for \( x\geq2 \) is a straight line. So we can use the equation of the line. Let's find two points on the line. We see that the line passes through \( (2,0) \) and \( (8,6) \) (since from \( x = 2 \) to \( x = 8 \), x increases by 6, y increases by 6, so slope \( m = 1 \)). So the equation is \( y-0 = 1\times(x - 2) \), so \( y=x - 2 \). Now, substitute \( x = 7 \) into the equation: \( y=7 - 2=5 \). Wait, but let's check the graph. When \( x = 7 \), on the line for \( x\geq2 \), the y - value should be 5? Wait, maybe. Alternatively, maybe I made a mistake in the slope. Wait, let's check the graph again. The line starts at \( (2,0) \) and goes up. So for each unit increase in x, y increases by 1. So when x = 2, y = 0; x = 3, y = 1; x = 4, y = 2; x = 5, y = 3; x = 6, y = 4; x = 7, y = 5. Yes, that makes sense. So \( f(7) \) is the y - value when \( x = 7 \) on the graph of \( y = f(x) \). Since for \( x\geq2 \), the function is a line with equation \( y=x - 2 \), so when \( x = 7 \), \( y=7 - 2 = 5 \).

Step2: Confirm the value

By substituting \( x = 7 \) into the equation of the line \( y=x - 2 \) (which represents the function for \( x\geq2 \)), we get \( y = 7-2=5 \). So \( f(7)=5 \).

Answer:

\( 5 \)