Question

find the value of $f(-7)$. $y = f(x)$

Explanation:

Step1: Identify the line for \( x = -7 \)

The left - hand part of the graph (for \( x \leq 2 \) approximately, from the graph's shape) is a straight line. We need to find the \( y \) - value when \( x=-7 \) on this line.
First, let's find the equation of the left - hand line. We can see that the line passes through the origin \((0,0)\) and let's take another point. When \( x=-2 \), let's assume the slope. Wait, actually, we can use the fact that the line goes from the left - hand side. Let's look at the slope. The line passes through \((0,0)\) and let's see the direction. Wait, when \( x = - 2\), what's the \( y \) - value? Wait, no, let's take two points on the left - hand line. Let's take \( x = 0,y = 0\) and another point. Wait, when \( x=-2\), maybe? Wait, no, the left - hand line: let's see, when \( x=-2\), what's the \( y \) - value? Wait, actually, the left - hand line has a slope. Let's calculate the slope \( m\) between two points. Let's take \((0,0)\) and \((-2,2)\)? Wait, no, looking at the graph, when \( x=-7\), we can see that the line from the left (the one with the arrow going up to the left) has a slope. Wait, actually, the left - hand line: let's see, when \( x = - 7\), we can find the \( y \) - value by looking at the graph's grid. Each grid square seems to be 1 unit. The line passes through \((0,0)\) and has a slope of \( m = 1\) (since for every 1 unit we move left in \( x\), we move up 1 unit in \( y\)). Wait, if \( x=-7\), then \( y = 7\)? Wait, no, wait. Wait, the line: when \( x=-1\), \( y = 1\); when \( x=-2\), \( y = 2\), so the equation of the left - hand line is \( y=x\) (because it passes through the origin and has a slope of 1). So when \( x=-7\), we substitute \( x = - 7\) into \( y=x\)? Wait, no, that can't be. Wait, maybe I got the slope wrong. Wait, the line is going from the upper left to the origin. Wait, when \( x=-7\), let's look at the graph. The left - hand line: if we take two points, say \((0,0)\) and \((-2,2)\), the slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{-2-0}=- 1\). Wait, that's a negative slope. Wait, when \( x=-2\), \( y = 2\)? No, maybe I messed up the direction. Wait, the arrow is going up to the left, so as \( x\) decreases (becomes more negative), \( y\) increases. So the slope \( m=\frac{\Delta y}{\Delta x}\). If \( x\) goes from 0 to - 2 (a change of \(\Delta x=-2\)), \( y\) goes from 0 to 2 (a change of \(\Delta y = 2\)), so \( m=\frac{2-0}{-2 - 0}=-1\). So the equation of the line is \( y=-x\) (because using the point - slope form \( y - y_1=m(x - x_1)\), with \((x_1,y_1)=(0,0)\) and \( m=-1\), we get \( y=-x\)). So when \( x=-7\), \( y=-(-7)=7\). Wait, let's check with the graph. If \( x=-7\), on the graph, the \( y \) - coordinate should be 7. Let's verify: the line \( y = - x\) passes through \((0,0)\), \((-1,1)\), \((-2,2)\), etc. So when \( x=-7\), \( y = 7\).

Step2: Confirm the value

By looking at the graph, the left - hand line (the one with the arrow going up to the left) has the equation \( y=-x\) (since it passes through the origin and has a slope of - 1). Substituting \( x = - 7\) into the equation \( y=-x\), we get \( y=-(-7)=7\).

Answer:

\( 7 \)