Question

use the graph of g(x), k(x) = 4x - 5, and h(x) = \\(\frac{x}{10}\\) to evaluate each of the following. a) g(8) = 1 b) k(2) + h(20) = c) g(2) - k(3) = d) x, if g(x) = -1

Explanation:

Part (a)

Step 1: Analyze the graph of \( g(x) \) at \( x = 8 \)

Looking at the graph of \( g(x) \), when \( x = 8 \), the \( y \)-value (which is \( g(8) \)) is 1. So \( g(8)=1 \).

Part (b)

Step 1: Evaluate \( k(2) \)

Given \( k(x)=4x - 5 \), substitute \( x = 2 \):
\( k(2)=4(2)-5=8 - 5 = 3 \)

Step 2: Evaluate \( h(20) \)

Given \( h(x)=\frac{x}{10} \), substitute \( x = 20 \):
\( h(20)=\frac{20}{10}=2 \)

Step 3: Add the results

\( k(2)+h(20)=3 + 2=5 \)

Part (c)

Step 1: Evaluate \( g(2) \) from the graph

From the graph of \( g(x) \), when \( x = 2 \), \( g(2) = 1 \).

Step 2: Evaluate \( k(3) \)

Given \( k(x)=4x - 5 \), substitute \( x = 3 \):
\( k(3)=4(3)-5=12 - 5 = 7 \)

Step 3: Subtract the results

\( g(2)-k(3)=1-7=-6 \)

Part (d)

Step 1: Find \( x \) when \( g(x)=-1 \)

Looking at the graph of \( g(x) \), we need to find the \( x \)-values where \( y=-1 \). From the graph, we can see that when \( y = - 1\), the \( x \)-values are \( x = 3 \) and \( x = 7 \)? Wait, no, let's re - examine the graph. Wait, the graph of \( g(x) \): let's check the points. Wait, the graph has a segment from \( x = 2 \) to \( x = 4 \) (descending) and from \( x = 4 \) to \( x = 8 \) (ascending). Wait, when does \( g(x)=-1 \)? Let's see the \( y \)-value of - 1. Looking at the grid, when \( y=-1 \), the \( x \)-coordinates are \( x = 3 \) and \( x = 7 \)? Wait, no, let's check the graph again. Wait, the graph: at \( x = 3 \), what's \( g(3) \)? Wait, the graph has a vertex at \( (4,-3) \). The line from \( (2,1) \) to \( (4,-3) \): the slope is \( \frac{-3 - 1}{4 - 2}=\frac{-4}{2}=-2 \). The equation of that line is \( y-1=-2(x - 2) \), \( y=-2x + 5 \). When \( y=-1 \), \( -1=-2x + 5 \), \( -2x=-6 \), \( x = 3 \). The line from \( (4,-3) \) to \( (8,1) \): slope is \( \frac{1+3}{8 - 4}=\frac{4}{4}=1 \). Equation: \( y + 3=1(x - 4) \), \( y=x - 7 \). When \( y=-1 \), \( -1=x - 7 \), \( x = 6 \)? Wait, maybe I made a mistake. Wait, the graph: looking at the given graph, the horizontal lines are at \( y = 1,y = 0,y=-1,y=-2,y=-3 \). The graph of \( g(x) \): from \( x=-\infty \) to \( x = 2 \), it's a horizontal line at \( y = 1 \). Then from \( x = 2 \) to \( x = 4 \), it's a line going down to \( (4,-3) \), then from \( x = 4 \) to \( x = 8 \), it's a line going up to \( (8,1) \), then horizontal to \( \infty \) at \( y = 1 \). So when \( y=-1 \), let's solve for \( x \) in the two line segments.

First segment: \( x\in[2,4] \), \( y=-2x + 5 \) (as above). Set \( y=-1 \): \( -1=-2x + 5\Rightarrow2x = 6\Rightarrow x = 3 \).

Second segment: \( x\in[4,8] \), \( y=x - 7 \) (since from \( (4,-3) \), slope 1: \( y+3=1(x - 4)\Rightarrow y=x - 7 \)). Set \( y=-1 \): \( -1=x - 7\Rightarrow x = 6 \). Wait, but maybe the problem expects the values? Wait, the graph: when \( g(x)=-1 \), the \( x \)-values are \( x = 3 \) and \( x = 6 \)? Wait, no, let's check the original graph again. The user's graph: the red graph. At \( x = 3 \), what's the \( y \)-value? Let's see the grid. The \( y \)-axis has marks at 2,1,0, - 1, - 2, - 3. The \( x \)-axis at 1,2,3,4,5,6,7,8,9. The graph: from \( x = 0 \) to \( x = 2 \), it's a horizontal line at \( y = 1 \). Then from \( x = 2 \) to \( x = 4 \), it goes down to \( (4,-3) \). Then from \( x = 4 \) to \( x = 8 \), it goes up to \( (8,1) \). Then horizontal. So when \( y=-1 \), on the line from \( (2,1) \) to \( (4,-3) \): when \( x = 3 \), \( y=-1 \) (since from \( x = 2 \) (y=1) to \( x = 4 \) (y=-3), the change in \( x \) is 2, change in \( y \) is - 4, so per unit \( x \), change in \( y \) is - 2. So at \( x = 2 + 1=3 \), \( y=1-2= - 1 \). On the line from \( (4,-3) \) to \( (8,1) \): change in \( x \) is 4, change in \( y \) is 4, so per unit \( x \), change in \( y \) is 1. So at \( x = 4+2=6 \), \( y=-3 + 2=-1 \). So the solutions for \( g(x)=-1 \) are \( x = 3 \) and \( x = 6 \). But maybe the problem expects both? Wait, the question says "x, if \( g(x)=-1 \)". So the values of \( x \) are \( x = 3 \) and \( x = 6 \).

Answer:

a) \( \boldsymbol{1} \)

b) \( \boldsymbol{5} \)

c) \( \boldsymbol{-6} \)

d) \( \boldsymbol{3} \) and \( \boldsymbol{6} \) (If we consider the graph, the \( x \)-values where \( g(x)=-1 \) are \( x = 3 \) and \( x = 6 \))