given the functions f(x), the piecewise defined function and g(x), the straight line, shown, answer the following questions. show all work 7/8
7. find f(2)
8. find g(-2)
9. find (f + g)(-1)
10. find (f - g)(1)

Question

Accepted Answer

### Question 7: Find $ f(2) $
# Explanation:
## Step 1: Analyze the graph of $ f(x) $
The function $ f(x) $ is a piecewise - defined function (the green graph). We need to find the $ y $ - value of $ f(x) $ when $ x = 2 $. By looking at the graph of $ f(x) $, we can see the coordinates of the points on the graph. For $ x=2 $, we locate the point on the $ f(x) $ graph with $ x $ - coordinate 2. From the graph, we observe that when $ x = 2 $, the $ y $ - coordinate (the value of $ f(x) $) is 4.
# Answer:
$ f(2)=4 $

### Question 8: Find $ g(-2) $
# Explanation:
## Step 1: Analyze the graph of $ g(x) $
The function $ g(x) $ is a straight - line (the blue graph). The equation of a straight line in slope - intercept form is $ y=mx + b $, where $ m $ is the slope and $ b $ is the $ y $ - intercept. First, we find the slope of $ g(x) $. Let's take two points on $ g(x) $. From the graph, we can see that the line passes through $ (0,0) $ and $ ( - 8,8) $ (or other pairs of points). The slope $ m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 8}{0-( - 8)}=\frac{- 8}{8}=- 1 $. The $ y $ - intercept $ b = 0 $ (since it passes through the origin $ (0,0) $). So the equation of $ g(x) $ is $ g(x)=-x $.
## Step 2: Substitute $ x=-2 $ into $ g(x) $
Substitute $ x = - 2 $ into $ g(x)=-x $. Then $ g(-2)=-(-2)=2 $. We can also verify this by looking at the graph: when $ x=-2 $, we find the point on the $ g(x) $ line with $ x $ - coordinate - 2, and the $ y $ - coordinate is 2.
# Answer:
$ g(-2)=2 $

### Question 9: Find $ (f + g)(-1) $
# Explanation:
## Step 1: Recall the definition of function addition
By the definition of function addition, $ (f + g)(x)=f(x)+g(x) $. So $ (f + g)(-1)=f(-1)+g(-1) $.
## Step 2: Find $ f(-1) $
Looking at the graph of $ f(x) $ (the green graph), when $ x=-1 $, we find the $ y $ - value of $ f(x) $. From the graph, when $ x = - 1 $, $ f(-1)=1 $.
## Step 3: Find $ g(-1) $
Using the equation of $ g(x)=-x $ (from question 8), substitute $ x=-1 $ into $ g(x) $. We get $ g(-1)=-(-1) = 1 $.
## Step 4: Calculate $ (f + g)(-1) $
$ (f + g)(-1)=f(-1)+g(-1)=1 + 1=2 $.
# Answer:
$ (f + g)(-1)=2 $

### Question 10: Find $ (f - g)(1) $
# Explanation:
## Step 1: Recall the definition of function subtraction
By the definition of function subtraction, $ (f - g)(x)=f(x)-g(x) $. So $ (f - g)(1)=f(1)-g(1) $.
## Step 2: Find $ f(1) $
Looking at the graph of $ f(x) $ (the green graph), when $ x = 1 $, we find the $ y $ - value of $ f(x) $. From the graph, when $ x = 1 $, $ f(1)=3 $.
## Step 3: Find $ g(1) $
Using the equation of $ g(x)=-x $, substitute $ x = 1 $ into $ g(x) $. We get $ g(1)=-1 $.
## Step 4: Calculate $ (f - g)(1) $
$ (f - g)(1)=f(1)-g(1)=3-(-1)=3 + 1=4 $.
# Answer:
$ (f - g)(1)=4 $

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QUESTION IMAGE

Question

Explanation:

Question 7: Find \( f(2) \)

Step 1: Analyze the graph of \( f(x) \)

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

Step 2: Substitute \( x=-2 \) into \( g(x) \)

Step 1: Recall the definition of function addition

Step 2: Find \( f(-1) \)

Step 3: Find \( g(-1) \)

Step 4: Calculate \( (f + g)(-1) \)

Answer:

Question 8: Find \( g(-2) \)