Question

Use the graph of f and g to evaluate the functions.
a. ​(fplus​g)(negative 6​) b.​ (fcircle​g)(5​)
c. left parenthesis StartFraction f Over g EndFraction right parenthesis
​(negative 2​) d.​ (gcircle​f)(2​)
Question content area bottom left
Part 1
-14
-12
-10
-8
-6
-4
-2
2
4
6
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10
12
14
-14
-12
-10
-8
-6
-4
-2
2
4
6
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14
x
y
y equals f left parenthesis x right parenthesisy equals g left parenthesis x right parenthesis



A coordinate system has a horizontal x-axis labeled from negative 14 to 14 in increments of 1 and a vertical y-axis labeled from negative 14 to 14 in increments of 1. A V-shaped graph labeled y = f(x) that opens upward has vertex (3, 0) and passes through the points (2, 1) and (4, 1). A line labeled y = g(x) falls from left to right and passes through the points (negative 1, 0) and (0, negative 1).
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Question content area right
Part 1
a. ​(fplus​g)(negative 6​)equals
  
enter your response here
​(Simplify your​ answer.)
Part 2
b.​ (fcircle​g)(5​)equals
  
enter your response here
​(Simplify your​ answer.)
Part 3
c. left parenthesis StartFraction f Over g EndFraction right parenthesis
​(negative 2​)equals
  
enter your response here
​(Simplify your​ answer.)
Part 4
d.​ (gcircle​f)(2​)equals
  
enter your response here

Answer 1

To evaluate the functions \( f \) and \( g \) at the specified inputs, we will need to analyze the provided descriptions.

Given:
- The function \( f(x) \) is a V-shaped graph centered at \( (3, 0) \). Its vertex is at \( (3, 0) \), and it passes through \( (2, 1) \) and \( (4, 1) \).
- The function \( g(x) \) is a line that falls from left to right, passing through the points \( (-1, 0) \) and \( (0, -1) \).

### Part 1: Evaluate \( (f + g)(-6) \)

To find \( (f + g)(-6) \):
1. First, we need \( f(-6) \) and \( g(-6) \).

- For \( f(-6) \): Given that it forms a V-shape with its vertex at \( (3, 0) \), the function \( f(x) \) can be calculated based on its symmetry:
\[
f(x) = |x - 3|
\]
Thus,
\[
f(-6) = |-6 - 3| = |-9| = 9
\]

- For \( g(-6) \): The line \( g(x) \) can be expressed as:
\[
g(x) = -x - 1
\]
So,
\[
g(-6) = -(-6) - 1 = 6 - 1 = 5
\]

Now, combine the results:
\[
(f + g)(-6) = f(-6) + g(-6) = 9 + 5 = 14
\]

### Part 2: Evaluate \( (f \circ g)(5) \)

To find \( (f \circ g)(5) \):
1. First evaluate \( g(5) \).
2. Then evaluate \( f(g(5)) \).

- For \( g(5) \):
\[
g(5) = -5 - 1 = -6
\]

- Next, find \( f(-6) \):
From previous computation, \( f(-6) = 9 \).

Thus,
\[
(f \circ g)(5) = f(g(5)) = f(-6) = 9
\]

### Part 3: Evaluate \( \left( \frac{f}{g} \right)(-2) \)

To find \( \left( \frac{f}{g} \right)(-2) \):
1. Evaluate \( f(-2) \) and \( g(-2) \).

- For \( f(-2) \):
\[
f(-2) = |-2 - 3| = |-5| = 5
\]

- For \( g(-2) \):
\[
g(-2) = -(-2) - 1 = 2 - 1 = 1
\]

Now, compute:
\[
\left( \frac{f}{g} \right)(-2) = \frac{f(-2)}{g(-2)} = \frac{5}{1} = 5
\]

### Part 4: Evaluate \( (g \circ f)(2) \)

To find \( (g \circ f)(2) \):
1. Evaluate \( f(2) \).
2. Then evaluate \( g(f(2)) \).

- For \( f(2) \):
\[
f(2) = |2 - 3| = |-1| = 1
\]

- Next, find \( g(1) \):
\[
g(1) = -1 - 1 = -2
\]

Thus,
\[
(g \circ f)(2) = g(f(2)) = g(1) = -2
\]

### Summary of Results
- a. \( (f + g)(-6) = 14 \)
- b. \( (f \circ g)(5) = 9 \)
- c. \( \left( \frac{f}{g} \right)(-2) = 5 \)
- d. \( (g \circ f)(2) = -2 \)