Question

question
find ( h(5) ).
(graph of ( f(x) ) as a line on coordinate plane)
answer
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Explanation:

Step1: Identify the function's equation

The graph is a straight line, so it's a linear function. Let's find the slope \( m \) and y-intercept \( b \). From the graph, the y-intercept \( b = 5 \) (crosses y-axis at \( (0,5) \)). The line passes through \( (3,0) \). Using slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), \( m=\frac{0 - 5}{3 - 0}=-\frac{5}{3} \)? Wait, no, wait. Wait, when \( x = 0 \), \( y = 5 \); when \( x = 2.5 \)? Wait, no, the line crosses x-axis at \( x = 2.5 \)? Wait, no, looking at the graph, the line goes from \( (0,5) \) to \( (2.5,0) \)? Wait, no, the grid: each square is 1 unit. So from \( (0,5) \), when \( x = 2 \), \( y = 5 - 2m \), and when \( x = 3 \), \( y = 0 \). Wait, let's recalculate. Let's take two points: \( (0,5) \) and \( (2.5,0) \)? No, the x-intercept is at \( x = 2.5 \)? Wait, no, the graph shows the line crossing x-axis at \( x = 2.5 \)? Wait, no, the grid lines: x-axis from -10 to 10, y-axis from -10 to 10. The line starts at top left, crosses y-axis at \( (0,5) \), then goes down to cross x-axis at \( (2.5, 0) \)? Wait, no, the arrow: when x increases, y decreases. Let's use two points: \( (0,5) \) and \( (3,0) \)? Wait, no, at x=3, y=0? Wait, the graph: the line passes through (0,5) and (2.5, 0)? Wait, maybe I made a mistake. Wait, let's find the equation of the line. The general form is \( y = mx + b \). We know \( b = 5 \) (since it crosses y-axis at (0,5)). Now, find the slope. Let's take another point: when x=2, what's y? Let's see, from (0,5), moving 2 units right, how much down? The slope: let's take (0,5) and (2.5, 0). So slope \( m = \frac{0 - 5}{2.5 - 0} = -2 \). Wait, that makes sense. Because if m = -2, then the equation is \( y = -2x + 5 \). Let's check: when x=0, y=5 (correct). When x=2.5, y = -2*(2.5) + 5 = -5 + 5 = 0 (correct). So the function is \( h(x) = -2x + 5 \) (assuming h(x) is the same as f(x) in the graph, maybe a typo, h(x) is f(x)).

Step2: Calculate h(5)

Now, substitute \( x = 5 \) into the function \( h(x) = -2x + 5 \). So \( h(5) = -2*(5) + 5 = -10 + 5 = -5 \). Wait, but let's check the graph. When x=5, what's y? Let's go to x=5 on the x-axis, then go up/down to the line. The line at x=5: since the slope is -2, from x=0 (y=5), each x increases by 1, y decreases by 2. So at x=1, y=3; x=2, y=1; x=3, y=-1; x=4, y=-3; x=5, y=-5. Yes, that matches. So h(5) = -5.

Answer:

-5