Question

find the value of $f(9)$.
$y = f(x)$

Explanation:

Step1: Analyze the graph for \( x = 9 \)

We need to find the \( y \)-value (which is \( f(x) \)) when \( x = 9 \). Looking at the graph, the part of the function for \( x \geq 0 \) (or the right - hand part) seems to be a linear function. Let's assume the equation of the line for \( x\geq0 \). We can see that the line passes through the origin \((0,0)\) and we can find its slope. Let's take another point, for example, when \( x = 1 \), let's see the \( y \)-value. From the graph, when \( x = 1 \), \( y = 1 \) (since it passes through \((1,1)\)). The slope \( m\) of a line passing through \((x_1,y_1)\) and \((x_2,y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1}\). For the line passing through \((0,0)\) and \((1,1)\), the slope \( m=\frac{1 - 0}{1 - 0}=1\). So the equation of the line is \( y=x\) (using the slope - intercept form \( y=mx + b\), where \( b = 0\) as it passes through the origin).

Step2: Substitute \( x = 9 \) into the equation

Since the equation of the line for \( x\geq0 \) is \( y = x\), when \( x = 9\), we substitute \( x = 9\) into \( y=x\). So \( y=f(9)=9\).

(corrected):

Step1: Determine the equation of the right - hand line

We identify two points on the right - hand segment of the graph: \((0,-1)\) and \((1,0)\). The slope \( m\) of a line passing through \((x_1,y_1)=(0,-1)\) and \((x_2,y_2)=(1,0)\) is calculated as \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0-(-1)}{1 - 0}=1\). Using the slope - intercept form \( y=mx + b\), where \( m = 1\) and \( b=-1\) (since when \( x = 0\), \( y=-1\)), the equation of the line is \( y=x - 1\).

Step2: Substitute \( x = 9\) into the equation

Substitute \( x = 9\) into \( y=x - 1\). We get \( y=f(9)=9 - 1=8\).

Answer:

\( 8 \) (Wait, there is a mistake in the previous step. Let's re - examine the graph. Let's find two points on the right - hand line. Let's take \( x = 0\), \( y=- 1\)? No, wait, looking at the graph again. Wait, the line on the right: when \( x = 0\), \( y=-1\)? Wait, no, maybe my initial assumption was wrong. Let's look at the grid. Each square is 1 unit. Let's find the equation of the right - hand line. Let's take two points: when \( x = 1\), what's \( y\)? Wait, maybe the line passes through \((0,-1)\) and \((1,0)\)? Wait, no, let's count the slope. From \( x = 0\) to \( x = 1\), how much does \( y\) change? Wait, maybe the correct way is to look at the graph for \( x = 9\). Let's see the right - hand line: starting from the vertex (the lowest point) and going up. Wait, maybe the line has a slope of 1, but let's check the \( y\) - intercept. Wait, when \( x = 0\), the \( y\) - value is - 1? No, maybe I misread the graph. Wait, let's look at the graph again. The right - hand line: when \( x = 0\), \( y=-1\)? No, wait, the graph: the line on the right, let's take \( x = 8\), what's \( y\)? Wait, maybe the slope is 1, but let's check the point. Wait, maybe the correct answer is 8? Wait, no, let's do it properly. Let's find the equation of the right - hand segment. Let's find two points on the right - hand line. Let's take \( x = 0\), \( y=-1\) (from the graph, the line crosses the \( y\) - axis at \( y=-1\)) and \( x = 1\), \( y = 0\). Then the slope \( m=\frac{0-(-1)}{1 - 0}=1\). So the equation is \( y=x - 1\). Wait, that makes more sense. Let's check: when \( x = 0\), \( y=-1\) (correct). When \( x = 1\), \( y=1 - 1=0\) (correct). When \( x = 2\), \( y=2 - 1 = 1\) (correct). So the equation of the right - hand line is \( y=x - 1\)? No, wait, when \( x = 0\), \( y=-1\), \( x = 1\), \( y = 0\), \( x = 2\), \( y = 1\), \( x = 3\), \( y = 2\),..., \( x = 9\), \( y=9 - 1=8\). Ah, that's the mistake. The \( y\) - intercept is - 1, not 0. So the equation is \( y=x - 1\)? Wait, no, let's check the graph again. Wait, the line passes through \((0,-1)\) and \((1,0)\), so the slope is \( m=\frac{0 - (-1)}{1-0}=1\), so the equation is \( y=x - 1\). Then when \( x = 9\), \( y=9 - 1 = 8\). So the correct answer is 8.