let f be a continuous function with selected values in the table above. let g be the inverse of f, such that g(x) = f⁻¹(x). find the following values if possible.
1) f(f(2))
2) g(11)
3) g(9)
4) g(g(9))
5) (f⁻¹ ∘ f)(5)
6) f⁻¹(2)
the function h is defined over the interval -4 ≤ x ≤ 11 as shown above. let h⁻¹ represent the inverse of h.
7) what is the maximum value of h(x)?
8) what is the maximum value of h⁻¹(x)?
9) find h⁻¹(5)
10) find h⁻¹(-2)
11) find h⁻¹(h⁻¹(-1))

Question

Accepted Answer

### 1) $ f(f(2)) $
# Explanation:
## Step1: Find $ f(2) $ from the table
From the table, when $ x = 2 $, $ f(2)=9 $.
## Step2: Find $ f(9) $ from the table
From the table, when $ x = 9 $, $ f(9)=15 $.
# Answer:
$ 15 $

### 2) $ g(11) $ (where $ g(x)=f^{-1}(x) $)
# Explanation:
## Step1: Recall the definition of inverse function
$ g(11)=f^{-1}(11) $ means we need to find $ x $ such that $ f(x)=11 $.
## Step2: Find $ x $ from the table
From the table, when $ f(x)=11 $, $ x = 5 $.
# Answer:
$ 5 $

### 3) $ g(9) $ (where $ g(x)=f^{-1}(x) $)
# Explanation:
## Step1: Recall the definition of inverse function
$ g(9)=f^{-1}(9) $ means we need to find $ x $ such that $ f(x)=9 $.
## Step2: Find $ x $ from the table
From the table, when $ f(x)=9 $, $ x = 2 $.
# Answer:
$ 2 $

### 4) $ g(g(9)) $ (where $ g(x)=f^{-1}(x) $)
# Explanation:
## Step1: Find $ g(9) $
From part 3, $ g(9)=2 $.
## Step2: Find $ g(2) $
$ g(2)=f^{-1}(2) $, so we need to find $ x $ such that $ f(x)=2 $. From the table, when $ f(x)=2 $, $ x=-1 $.
# Answer:
$ -1 $

### 5) $ (f^{-1}\circ f)(5) $
# Explanation:
## Step1: Recall the property of inverse functions
$ (f^{-1}\circ f)(x)=x $ for all $ x $ in the domain of $ f $.
## Step2: Apply the property
So $ (f^{-1}\circ f)(5)=5 $.
# Answer:
$ 5 $

### 6) $ f^{-1}(2) $
# Explanation:
## Step1: Recall the definition of inverse function
$ f^{-1}(2) $ means we need to find $ x $ such that $ f(x)=2 $.
## Step2: Find $ x $ from the table
From the table, when $ f(x)=2 $, $ x=-1 $.
# Answer:
$ -1 $

### 7) What is the maximum value of $ h(x) $?
# Explanation:
## Step1: Analyze the graph of $ h $
The graph of $ h $ has its highest point (maximum) at the left - most point. From the graph, when $ x=-4 $, the $ y $ - value (which is $ h(x) $) is $ 7 $ (by looking at the $ y $ - coordinate of the left - most point of the graph).
# Answer:
$ 7 $

### 8) What is the maximum value of $ h^{-1}(x) $?
# Explanation:
## Step1: Recall the property of inverse functions and domain/range
The domain of $ h $ is $ - 4\leq x\leq11 $ and the range of $ h $ is the set of $ y $ - values of $ h(x) $. The inverse function $ h^{-1}(x) $ has a domain equal to the range of $ h $ and a range equal to the domain of $ h $. The maximum value of $ h^{-1}(x) $ is the maximum value in the domain of $ h $ (since the range of $ h^{-1} $ is the domain of $ h $).
## Step2: Determine the maximum value
The domain of $ h $ is $ -4\leq x\leq11 $, so the maximum value of $ h^{-1}(x) $ is $ 11 $.
# Answer:
$ 11 $

### 9) Find $ h^{-1}(5) $
# Explanation:
## Step1: Recall the definition of inverse function
$ h^{-1}(5) $ means we need to find $ x $ such that $ h(x)=5 $.
## Step2: Analyze the graph of $ h $
Looking at the graph of $ h $, we find the $ x $ - value for which $ h(x)=5 $. By looking at the graph, when $ h(x)=5 $, $ x = 0 $.
# Answer:
$ 0 $

### 10) Find $ h^{-1}(-2) $
# Explanation:
## Step1: Recall the definition of inverse function
$ h^{-1}(-2) $ means we need to find $ x $ such that $ h(x)=-2 $.
## Step2: Analyze the graph of $ h $
Looking at the graph of $ h $, we find the $ x $ - value for which $ h(x)=-2 $. By looking at the graph, when $ h(x)=-2 $, $ x = 6 $.
# Answer:
$ 6 $

### 11) Find $ h^{-1}(h^{-1}(-1)) $
# Explanation:
## Step1: Find $ h^{-1}(-1) $
First, we need to find $ x $ such that $ h(x)=-1 $. From the graph of $ h $, when $ h(x)=-1 $, $ x = 5 $. So $ h^{-1}(-1)=5 $.
## Step2: Find $ h^{-1}(5) $
Now we need to find $ x $ such that $ h(x)=5 $. From part 9, we know that when $ h(x)=5 $, $ x = 0 $. So $ h^{-1}(h^{-1}(-1))=h^{-1}(5)=0 $.
# Answer:
$ 0 $

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QUESTION IMAGE

Question

Explanation:

1) \( f(f(2)) \)

Step1: Find \( f(2) \) from the table

Step2: Find \( f(9) \) from the table

Step1: Recall the definition of inverse function

Step2: Find \( x \) from the table

Step1: Recall the definition of inverse function

Step2: Find \( x \) from the table

Answer:

2) \( g(11) \) (where \( g(x)=f^{-1}(x) \))