Question

find the value of $f(-5)$. $y = f(x)$

Explanation:

Step1: Identify the line equation

The left - hand part of the graph (for \(x\leq0\) approximately) is a straight line. We can find the equation of this line. First, we know two points on this line: the \(y\) - intercept is \((0, - 2)\) (when \(x = 0\), \(y=-2\)) and the \(x\) - intercept. From the graph, when \(y = 0\), let's assume the \(x\) - intercept is \((-1,0)\)? Wait, no, let's re - examine. Wait, the line passes through \((0,-2)\) and let's take another point. Let's see, when \(x=-5\), we need to find \(y\). Let's find the slope \(m\) of the line. Let's take two points: \((0,-2)\) and \((-1,0)\)? Wait, no, when \(x = - 1\), does \(y = 0\)? Wait, from the graph, the line crosses the \(x\) - axis at \(x=-1\) (since when \(y = 0\), \(x=-1\)) and the \(y\) - axis at \(y = - 2\) (when \(x = 0\)). So the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take \((x_1,y_1)=(-1,0)\) and \((x_2,y_2)=(0,-2)\). Then \(m=\frac{-2 - 0}{0-(-1)}=\frac{-2}{1}=-2\). So the equation of the line is \(y-0=-2(x + 1)\) (using point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-1,0)\) and \(m=-2\)), which simplifies to \(y=-2x - 2\).

Step2: Substitute \(x=-5\) into the line equation

Now, we want to find \(f(-5)\), which is the value of \(y\) when \(x=-5\) on the line \(y = f(x)\) (the left - hand line). Substitute \(x=-5\) into \(y=-2x - 2\).
\(y=-2\times(-5)-2\)
\(y = 10-2\)
\(y = 8\)? Wait, that can't be right. Wait, maybe I made a mistake in the points. Wait, let's re - check the graph. Wait, the line passes through \((0,-2)\) and when \(x=-1\), \(y = 0\)? Wait, no, maybe the two points are \((0,-2)\) and \((1,-4)\)? No, that doesn't seem right. Wait, maybe the slope is calculated incorrectly. Wait, let's look at the graph again. The line goes from the second quadrant (upper left) to the fourth quadrant (lower right). Let's take two points: when \(x = 0\), \(y=-2\); when \(x=-1\), \(y = 0\). So the slope \(m=\frac{0-(-2)}{-1 - 0}=\frac{2}{-1}=-2\), that part is correct. Then the equation is \(y=-2x-2\). Now, when \(x=-5\), \(y=-2\times(-5)-2=10 - 2 = 8\)? Wait, but let's check with \(x=-5\). Wait, maybe the line is different. Wait, another approach: the line passes through \((0,-2)\) and let's see the direction. Wait, maybe the line has a slope of \(- 1\)? No, let's count the rise over run. From \((0,-2)\) to \((-2,0)\): rise is \(0-(-2)=2\), run is \(-2 - 0=-2\), so slope \(m=\frac{2}{-2}=-1\). Ah! Maybe I took the wrong \(x\) - intercept. Let's assume the line passes through \((0,-2)\) and \((-2,0)\). Then the slope \(m=\frac{0-(-2)}{-2-0}=\frac{2}{-2}=-1\). Then the equation of the line is \(y-(-2)=-1(x - 0)\), so \(y+2=-x\), or \(y=-x - 2\). Now, substitute \(x=-5\) into \(y=-x - 2\). Then \(y=-(-5)-2=5 - 2 = 3\)? No, that's not matching. Wait, maybe the correct two points are \((0,-2)\) and \((-1, - 1)\)? No, this is getting confusing. Wait, let's look at the graph again. The line that includes \(x=-5\) is the left - hand line. Let's see, when \(x=-5\), we can find the \(y\) - value by using the fact that the line has a slope. Wait, maybe the line passes through \((0,-2)\) and \((-2,0)\). So the slope is \(\frac{0 - (-2)}{-2-0}=\frac{2}{-2}=-1\). So the equation is \(y=-x - 2\). Then when \(x=-5\), \(y=-(-5)-2=5 - 2 = 3\)? No, that's not right. Wait, maybe I made a mistake in the intercepts. Wait, the \(y\) - intercept is \((0,-2)\), and let's take another point. Let's say when \(x=-5\), what's the \(y\) - value? Wait, maybe the line is \(y = x+3\)? No, that doesn't fit \((0,-2)\). Wait, perhaps the correct way is to look at the graph: the left - hand line (the one with the arrow going up to the left) passes through \((0,-2)\) and when \(x=-5\), we can see that the \(y\) - value is 8? Wait, no, let's use the two - point formula correctly. Let's take two points on the line: \((0,-2)\) and \((-1,0)\) (from the graph, when \(x=-1\), \(y = 0\)). Then the slope \(m=\frac{0-(-2)}{-1 - 0}=\frac{2}{-1}=-2\). So the equation is \(y=-2x - 2\). Now, substitute \(x=-5\): \(y=-2\times(-5)-2=10 - 2 = 8\). Wait, but when \(x=-1\), \(y=-2\times(-1)-2=2 - 2 = 0\), which matches the \(x\) - intercept. When \(x = 0\), \(y=-2\times0-2=-2\), which matches the \(y\) - intercept. So that seems correct. So \(f(-5)=8\)? Wait, but let's check the graph again. The left - hand line (the one with the arrow in the second quadrant) when \(x=-5\), which is far to the left of the \(y\) - axis, the \(y\) - value should be positive. So if \(x=-5\), using \(y=-2x - 2\), \(y = 10 - 2=8\). So that's the value.

Answer:

\(f(-5)=8\)