Use the graph of f and g to evaluate the functions.

Question

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​)
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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. \( (g \circ f)(2) = -2 \)