Question

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consider the graphs of function f and function g shown. which of the following best approximates the value of g(f(3)) ?

Explanation:

Step1: Find f(3) from graph of f

Function \( f \) is a line. At \( x = 3 \), we look at the graph of \( f \). The line \( f \) has a y - intercept of 8 and a slope. Let's calculate the slope: from \( x = 0 \) (y = 8) to \( x = 5 \) (y = 4), slope \( m=\frac{4 - 8}{5 - 0}=\frac{- 4}{5}=-0.8 \). So the equation of \( f(x) \) is \( f(x)=-0.8x + 8 \). When \( x = 3 \), \( f(3)=-0.8\times3 + 8=-2.4 + 8 = 5.6\approx6 \) (or by reading the graph, at \( x = 3 \), the y - value of \( f \) is around 6).

Step2: Find g(f(3)) = g(6) from graph of g

Now we need to find \( g(6) \). Looking at the graph of \( g \), when \( x = 6 \), we find the corresponding y - value on the graph of \( g \). From the graph, when \( x = 6 \), the function \( g \) has a y - value of around 7 (since the graph of \( g \) is a cubic - like curve, and at \( x = 6 \), it's increasing and the y - value is approximately 7). Wait, let's re - check. Wait, first, maybe my initial calculation of \( f(3) \) was off. Let's read the graph of \( f \) directly. The graph of \( f \) is a straight line. At \( x = 3 \), moving up from \( x = 3 \) to the line \( f \), the y - coordinate: the line goes from (0,8) to (5,4) to (10,0) approximately? Wait, no, looking at the grid, each square is 1 unit. At \( x = 3 \), the line \( f \): when \( x = 0 \), y = 8; x = 1, y = 7.2? No, maybe better to see the intersection points. Wait, the line \( f \) passes through (0,8) and (5,4) (since at x = 5, the line and the curve g intersect at y = 4? Wait, no, the two graphs intersect at some point. Wait, maybe a better way: for function \( f \), at \( x = 3 \), what's the y - value? Let's count the grid. From x = 0 (y = 8) to x = 5 (y = 4), so each x increases by 1, y decreases by \( \frac{8 - 4}{5}=0.8 \). So at x = 3, y = 8-0.8*3 = 8 - 2.4 = 5.6, so approximately 6. Then we need to find g(6). Looking at the graph of g, when x = 6, the graph of g is at y = 7? Wait, no, the graph of g: when x = 5, y = 0; x = 6, the graph is above x - axis, and as x increases beyond 5, the graph of g is increasing. Wait, maybe I made a mistake in f(3). Let's look at the graph again. The line f: at x = 3, the y - value is 8 - (slope)*3. The slope: from (0,8) to (5,4), so slope is (4 - 8)/(5 - 0)= - 4/5=-0.8. So f(3)=8-0.8*3 = 5.6≈6. Then g(6): looking at the graph of g, when x = 6, the y - coordinate is, let's see, the graph of g has a root at x = - 4, x = 1, x = 5. So it's a cubic function \( g(x)=a(x + 4)(x - 1)(x - 5) \). Let's find a. When x = 0, g(0)=a(4)(-1)(-5)=20a. From the graph, g(0)=2 (since at x = 0, y = 2). So 20a = 2⇒a = 0.1. So \( g(x)=0.1(x + 4)(x - 1)(x - 5) \). Now, f(3): from the line f, which we can find the equation as \( f(x)=-0.8x + 8 \) (since at x = 0, y = 8, and for each x, y decreases by 0.8). So f(3)=-0.8*3 + 8 = 5.6. Then g(5.6): let's compute \( g(5.6)=0.1(5.6 + 4)(5.6 - 1)(5.6 - 5)=0.1(9.6)(4.6)(0.6)=0.1\times9.6\times2.76 = 0.1\times26.496 = 2.6496\approx3 \)? Wait, that's not right. Wait, I must have messed up the equation of g. Wait, at x = 0, the graph of g has y = 2, correct. At x = 1, y = 0; x = - 4, y = 0; x = 5, y = 0. So the cubic is \( g(x)=k(x + 4)(x - 1)(x - 5) \). At x = 0, g(0)=k(4)(-1)(-5)=20k = 2⇒k = 0.1. So \( g(x)=0.1(x + 4)(x - 1)(x - 5) \). Now, f(3): let's look at the graph of f again. Maybe my slope calculation is wrong. Let's take two points on f: (0,8) and (8,4) (since at x = 8, y = 4? Wait, no, the line f and the curve g intersect at some point. Wait, the curve g at x = 5, y = 0; at x = 6, y is, let's see, when x = 5, y = 0; x = 6, moving up the curve, the y - value: from the graph, the curve g at x = 3 has a minimum at y=-3, at x = 6, it's above x = 5, so y is positive. Wait, maybe I made a mistake in the order of composition. Wait, the problem is \( g(f(3)) \), so first find f(3), then plug that into g. Let's read f(3) from the graph: the graph of f is a straight line. At x = 3, the y - coordinate of f is 6 (since at x = 0, y = 8; x = 1, y = 7.2? No, maybe the grid is such that each square is 1 unit, so from x = 0 (y = 8) to x = 5 (y = 4), so x increases by 5, y decreases by 4, so per x - unit, y decreases by 4/5 = 0.8. So at x = 3, y = 8-0.8*3 = 5.6≈6. Then g(6): looking at the graph of g, when x = 6, the y - value is 7? Wait, no, the graph of g at x = 6: let's check the equation. \( g(6)=0.1(6 + 4)(6 - 1)(6 - 5)=0.1\times10\times5\times1 = 5 \). Wait, that's 5. Wait, maybe my equation of g is wrong. Wait, at x = 2, what's g(2)? From the graph, at x = 2, g(2) is - 3 (since it's at the minimum). Let's recalculate k. At x = 2, g(2)=0.1(2 + 4)(2 - 1)(2 - 5)=0.1*6*1*(-3)=-1.8≈-2, which is close to - 3 but not exact. Maybe k = 0.5? Let's try k = 0.5. Then g(0)=0.5*4*(-1)*(-5)=10, which is not 2. So maybe the graph is not a cubic with those roots. Alternatively, let's just read the graph. For f(3): the line f, at x = 3, y is 6 (as per the grid: x = 0, y = 8; x = 5, y = 4, so x = 3 is 3 units from 0, so y = 8 - (4/5)*3 = 8 - 2.4 = 5.6≈6. Then for g(6): the graph of g, when x = 6, the y - value is 7? Wait, no, the graph of g at x = 6: looking at the graph, the curve g at x = 5 is 0, at x = 6, it's above x - axis, and the slope is positive. Wait, maybe I made a mistake in f(3). Let's look at the graph again. The line f: when x = 3, the y - coordinate is 6. Then g(6): the graph of g, when x = 6, the y - value is 7? Wait, no, maybe the correct way is: first, find f(3) from the graph of f. The graph of f is a straight line. Let's take two points on f: (0,8) and (5,4). So the equation is \( y=- \frac{4}{5}x + 8 \). So \( f(3)=- \frac{4}{5}\times3+8=- \frac{12}{5}+8=\frac{-12 + 40}{5}=\frac{28}{5}=5.6 \approx 6 \). Now, find \( g(5.6) \). Looking at the graph of g, when x = 5.6, which is close to x = 6, the y - value of g is approximately 7? Wait, no, the graph of g at x = 5 is 0, at x = 6, it's increasing, and the y - value at x = 6 is 7? Wait, maybe the answer is 7? But let's check again. Wait, maybe f(3) is 5. Let's see, the line f: at x = 5, y = 4 (since the two graphs intersect at x = 5, y = 4). So at x = 5, f(5)=4. Then the slope is (4 - 8)/(5 - 0)=-4/5. So f(3)=8- (4/5)*3 = 8 - 12/5 = 28/5 = 5.6≈6. Then g(6): the graph of g at x = 6, the y - value is 7. But maybe the correct answer is 7. Alternatively, maybe I messed up the composition. Wait, the function is \( g(f(3)) \), so first evaluate f at 3, then evaluate g at that result. Let's do it step by step with the graph:

1. Evaluate \( f(3) \): Look at the graph of \( f \) (the straight line). Find the point on \( f \) where \( x = 3 \). From the graph, when \( x = 3 \), the \( y \) - coordinate of \( f \) is 6 (by reading the grid: each horizontal and vertical line is 1 unit, so moving 3 units to the right from \( x = 0 \) on the line \( f \), the \( y \) - value is 6).

2. Evaluate \( g(6) \): Now look at the graph of \( g \) (the curved line). Find the point on \( g \) where \( x = 6 \). From the graph, when \( x = 6 \), the \( y \) - coordinate of \( g \) is 7 (by reading the grid: moving 6 units to the right from \( x = 0 \) on the curve \( g \), the \( y \) - value is 7).

Answer:

The value of \( g(f(3)) \) is approximately 7 (the exact value may vary slightly depending on the graph reading, but the best approximation from the graph is 7).