https://r2.erweima.ai/mathbot/image/1725843291770ek4jii50.jpg

Use the graphs of f and g to evaluate each expression. (If an answer is undefined, enter UNDEFINED.)

(a)    
(f ∘ g)(3)

 

(b)    
g(f(2))

 

(c)    
g(f(5))

 

(d)    
(f ∘ g)(−3)

 

(e)    
(g ∘ f)(−1)

 

(f)    
f(g(−1))


We are given a composite function problem with two graphs: \( f(x) \) and \( g(x) \). To evaluate the composite expressions, we will analyze both graphs to find the values of \( f(x) \) and \( g(x) \) at the specified points.

Here are the steps to solve each part:

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**(a) \( (f \circ g)(3) \)**

This means we need to find \( f(g(3)) \). First, we evaluate \( g(3) \) from the graph of \( g(x) \), then plug that result into \( f(x) \).

1. From the graph of \( g(x) \), locate \( g(3) \).
2. Use this value to evaluate \( f(g(3)) \).

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**(b) \( g(f(2)) \)**

This means we need to find \( g(f(2)) \). First, evaluate \( f(2) \) from the graph of \( f(x) \), then plug that result into \( g(x) \).

1. From the graph of \( f(x) \), locate \( f(2) \).
2. Use this value to evaluate \( g(f(2)) \).

---

**(c) \( g(f(5)) \)**

This means we need to find \( g(f(5)) \). First, evaluate \( f(5) \) from the graph of \( f(x) \), then plug that result into \( g(x) \).

1. From the graph of \( f(x) \), locate \( f(5) \).
2. Use this value to evaluate \( g(f(5)) \).

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**(d) \( (f \circ g)(-3) \)**

This means we need to find \( f(g(-3)) \). First, evaluate \( g(-3) \) from the graph of \( g(x) \), then plug that result into \( f(x) \).

1. From the graph of \( g(x) \), locate \( g(-3) \).
2. Use this value to evaluate \( f(g(-3)) \).

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**(e) \( (g \circ f)(-1) \)**

This means we need to find \( g(f(-1)) \). First, evaluate \( f(-1) \) from the graph of \( f(x) \), then plug that result into \( g(x) \).

1. From the graph of \( f(x) \), locate \( f(-1) \).
2. Use this value to evaluate \( g(f(-1)) \).

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**(f) \( f(g(-1)) \)**

This means we need to find \( f(g(-1)) \). First, evaluate \( g(-1) \) from the graph of \( g(x) \), then plug that result into \( f(x) \).

1. From the graph of \( g(x) \), locate \( g(-1) \).
2. Use this value to evaluate \( f(g(-1)) \).

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Let me analyze the graph and work out the exact values for each expression.I've displayed the image of the graph for further analysis. I will now extract values of \( f(x) \) and \( g(x) \) from the graph to solve the given problems.

Let's proceed with the evaluations:

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**(a) \( (f \circ g)(3) \)**

From the graph:
- \( g(3) = 4 \).
- Now, evaluate \( f(4) \). From the graph of \( f(x) \), \( f(4) = 8 \).

So, \( (f \circ g)(3) = 8 \).

---

**(b) \( g(f(2)) \)**

From the graph:
- \( f(2) = 2 \).
- Now, evaluate \( g(2) \). From the graph of \( g(x) \), \( g(2) = 2 \).

So, \( g(f(2)) = 2 \).

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**(c) \( g(f(5)) \)**

From the graph:
- \( f(5) \) is undefined since there's no point for \( f(x) \) at \( x = 5 \).

Thus, \( g(f(5)) = \text{UNDEFINED} \).

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**(d) \( (f \circ g)(-3) \)**

From the graph:
- \( g(-3) = -2 \).
- Now, evaluate \( f(-2) \). From the graph of \( f(x) \), \( f(-2) = 2 \).

So, \( (f \circ g)(-3) = 2 \).

---

**(e) \( (g \circ f)(-1) \)**

From the graph:
- \( f(-1) = 0 \).
- Now, evaluate \( g(0) \). From the graph of \( g(x) \), \( g(0) = -2 \).

So, \( (g \circ f)(-1) = -2 \).

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**(f) \( f(g(-1)) \)**

From the graph:
- \( g(-1) = 0 \).
- Now, evaluate \( f(0) \). From the graph of \( f(x) \), \( f(0) = -2 \).

So, \( f(g(-1)) = -2 \).

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### Final Answers:
- (a) \( (f \circ g)(3) = 8 \)
- (b) \( g(f(2)) = 2 \)
- (c) \( g(f(5)) = \text{UNDEFINED} \)
- (d) \( (f \circ g)(-3) = 2 \)
- (e) \( (g \circ f)(-1) = -2 \)
- (f) \( f(g(-1)) = -2 \)

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. How do you interpret the composite function \( f(g(x)) \) graphically?
2. What conditions cause a composite function to be undefined?
3. Can \( g(f(x)) \) give different results than \( f(g(x)) \)? Why or why not?
4. How do you identify the domain of a composite function using graphs?
5. What is the significance of zeroes in composite function evaluations?

**Tip:** Always ensure that the range of the inner function fits within the domain of the outer function when dealing with composite functions.

Use the graphs of f and g to evaluate each expression. (If an answer is undefined, enter UNDEFINED.)

(a) (f ∘ g)(3)
(b) g(f(2))
(c) g(f(5))
(d) (f ∘ g)(−3)
(e) (g ∘ f)(−1)
(f) f(g(−1))

Learn how to evaluate composite functions such as (f ∘ g)(3) and g(f(2)) using graphs. This exercise covers key concepts in composite function analysis and graph interpretation, suitable for Grades 10-12 students.

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