Answer :
Let's go through the problem one step at a time.
#### Given Information:
- Function [tex]\( f(x) = -2x + 5 \)[/tex]
- Values of function [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & 6 \\ \hline 0 & 1 \\ \hline 1 & -2 \\ \hline 4 & 7 \\ \hline 6 & 10 \\ \hline 7 & 6 \\ \hline \end{array} \][/tex]
Information about [tex]\( h(x) \)[/tex] is not provided, so we’ll answer the questions requiring this function based on the given data only.
#### 1) What is the value of [tex]\( f(4) \)[/tex]?
To find [tex]\( f(4) \)[/tex], substitute [tex]\( x = 4 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(4) = -2(4) + 5 = -8 + 5 = -3 \][/tex]
Answer: [tex]\( f(4) = -3 \)[/tex]
#### 2) What is the value of [tex]\( g(4) \)[/tex]?
From the table provided, when [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = 7 \][/tex]
Answer: [tex]\( g(4) = 7 \)[/tex]
#### 3) What is the value of [tex]\( h(4) \)[/tex]?
Insufficient information about the function [tex]\( h \)[/tex] is provided, so we cannot determine [tex]\( h(4) \)[/tex].
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 4-5) For what values of [tex]\( x \)[/tex] is [tex]\( f(x)=13 \)[/tex], [tex]\( g(x)=6 \)[/tex], [tex]\( h(x)=1 \)[/tex]?
##### 4) For what values of [tex]\( x \)[/tex] is [tex]\( f(x)=13 \)[/tex]?
To solve, set [tex]\( f(x) = 13 \)[/tex]:
[tex]\[ -2x + 5 = 13 \][/tex]
Subtracting 5 from both sides:
[tex]\[ -2x = 8 \][/tex]
Dividing both sides by -2:
[tex]\[ x = -4 \][/tex]
Answer: [tex]\( x = -4 \)[/tex]
##### 5) For what values of [tex]\( x \)[/tex] is [tex]\( g(x)=6 \)[/tex]?
From the table, [tex]\( g(x) = 6 \)[/tex] for [tex]\( x = -2 \)[/tex] and [tex]\( x = 7 \)[/tex].
Answer: [tex]\( x = -2 \)[/tex] or [tex]\( x = 7 \)[/tex]
##### 6) For what values of [tex]\( x \)[/tex] is [tex]\( h(x)=1 \)[/tex]?
Insufficient information about the function [tex]\( h \)[/tex] is provided, so we cannot determine values of [tex]\( x \)[/tex] for which [tex]\( h(x) = 1 \)[/tex].
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 7) What is the value of [tex]\( f(2) + g(7) + h(0) \)[/tex]?
Given [tex]\( h(0) \)[/tex] cannot be determined without information about [tex]\( h \)[/tex]. Hence, it's not possible to solve this part accurately.
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 8) What is the value of [tex]\( f(g(0)) \)[/tex]?
First, find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 1 \][/tex]
Next, find [tex]\( f(g(0)) = f(1) \)[/tex]:
[tex]\[ f(1) = -2(1) + 5 = -2 + 5 = 3 \][/tex]
Answer: [tex]\( f(g(0)) = 3 \)[/tex]
#### 9) What is the value of [tex]\( h(f(0)) \)[/tex]?
First, find [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = -2(0) + 5 = 5 \][/tex]
Next, find [tex]\( h(f(0)) = h(5) \)[/tex], but insufficient information about function [tex]\( h \)[/tex] is provided.
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 10) What is the value of [tex]\( f(h(g(-2))) \)[/tex]?
First, find [tex]\( g(-2) \)[/tex]:
[tex]\[ g(-2) = 6 \][/tex]
Next, find [tex]\( h(g(-2)) = h(6) \)[/tex], but insufficient information about [tex]\( h \)[/tex] is provided.
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 11-14) Membership in the domain and range of [tex]\( h \)[/tex]
##### 11) Is the number 6 a member of the domain of function [tex]\( h \)[/tex]?
Insufficient information about the domain of [tex]\( h \)[/tex] is provided.
Answer: Information about the domain of [tex]\( h(x) \)[/tex] is not provided.
##### 12) Is the number -5.5 a member of the domain of function [tex]\( h \)[/tex]?
Insufficient information about the domain of [tex]\( h \)[/tex] is provided.
Answer: Information about the domain of [tex]\( h(x) \)[/tex] is not provided.
##### 13) Is the number 6 a member of the range of function [tex]\( h \)[/tex]?
Insufficient information about the range of [tex]\( h \)[/tex] is provided.
Answer: Information about the range of [tex]\( h(x) \)[/tex] is not provided.
##### 14) Is the number -5.5 a member of the range of function [tex]\( h \)[/tex]?
Insufficient information about the range of [tex]\( h \)[/tex] is provided.
Answer: Information about the range of [tex]\( h(x) \)[/tex] is not provided.
It's important to note that the lack of information about [tex]\( h \)[/tex] makes it impossible to accurately answer several questions. If additional information about [tex]\( h \)[/tex] is provided, these questions could be revisited.
#### Given Information:
- Function [tex]\( f(x) = -2x + 5 \)[/tex]
- Values of function [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & 6 \\ \hline 0 & 1 \\ \hline 1 & -2 \\ \hline 4 & 7 \\ \hline 6 & 10 \\ \hline 7 & 6 \\ \hline \end{array} \][/tex]
Information about [tex]\( h(x) \)[/tex] is not provided, so we’ll answer the questions requiring this function based on the given data only.
#### 1) What is the value of [tex]\( f(4) \)[/tex]?
To find [tex]\( f(4) \)[/tex], substitute [tex]\( x = 4 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(4) = -2(4) + 5 = -8 + 5 = -3 \][/tex]
Answer: [tex]\( f(4) = -3 \)[/tex]
#### 2) What is the value of [tex]\( g(4) \)[/tex]?
From the table provided, when [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = 7 \][/tex]
Answer: [tex]\( g(4) = 7 \)[/tex]
#### 3) What is the value of [tex]\( h(4) \)[/tex]?
Insufficient information about the function [tex]\( h \)[/tex] is provided, so we cannot determine [tex]\( h(4) \)[/tex].
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 4-5) For what values of [tex]\( x \)[/tex] is [tex]\( f(x)=13 \)[/tex], [tex]\( g(x)=6 \)[/tex], [tex]\( h(x)=1 \)[/tex]?
##### 4) For what values of [tex]\( x \)[/tex] is [tex]\( f(x)=13 \)[/tex]?
To solve, set [tex]\( f(x) = 13 \)[/tex]:
[tex]\[ -2x + 5 = 13 \][/tex]
Subtracting 5 from both sides:
[tex]\[ -2x = 8 \][/tex]
Dividing both sides by -2:
[tex]\[ x = -4 \][/tex]
Answer: [tex]\( x = -4 \)[/tex]
##### 5) For what values of [tex]\( x \)[/tex] is [tex]\( g(x)=6 \)[/tex]?
From the table, [tex]\( g(x) = 6 \)[/tex] for [tex]\( x = -2 \)[/tex] and [tex]\( x = 7 \)[/tex].
Answer: [tex]\( x = -2 \)[/tex] or [tex]\( x = 7 \)[/tex]
##### 6) For what values of [tex]\( x \)[/tex] is [tex]\( h(x)=1 \)[/tex]?
Insufficient information about the function [tex]\( h \)[/tex] is provided, so we cannot determine values of [tex]\( x \)[/tex] for which [tex]\( h(x) = 1 \)[/tex].
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 7) What is the value of [tex]\( f(2) + g(7) + h(0) \)[/tex]?
Given [tex]\( h(0) \)[/tex] cannot be determined without information about [tex]\( h \)[/tex]. Hence, it's not possible to solve this part accurately.
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 8) What is the value of [tex]\( f(g(0)) \)[/tex]?
First, find [tex]\( g(0) \)[/tex]:
[tex]\[ g(0) = 1 \][/tex]
Next, find [tex]\( f(g(0)) = f(1) \)[/tex]:
[tex]\[ f(1) = -2(1) + 5 = -2 + 5 = 3 \][/tex]
Answer: [tex]\( f(g(0)) = 3 \)[/tex]
#### 9) What is the value of [tex]\( h(f(0)) \)[/tex]?
First, find [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = -2(0) + 5 = 5 \][/tex]
Next, find [tex]\( h(f(0)) = h(5) \)[/tex], but insufficient information about function [tex]\( h \)[/tex] is provided.
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 10) What is the value of [tex]\( f(h(g(-2))) \)[/tex]?
First, find [tex]\( g(-2) \)[/tex]:
[tex]\[ g(-2) = 6 \][/tex]
Next, find [tex]\( h(g(-2)) = h(6) \)[/tex], but insufficient information about [tex]\( h \)[/tex] is provided.
Answer: Information about [tex]\( h(x) \)[/tex] is not provided.
#### 11-14) Membership in the domain and range of [tex]\( h \)[/tex]
##### 11) Is the number 6 a member of the domain of function [tex]\( h \)[/tex]?
Insufficient information about the domain of [tex]\( h \)[/tex] is provided.
Answer: Information about the domain of [tex]\( h(x) \)[/tex] is not provided.
##### 12) Is the number -5.5 a member of the domain of function [tex]\( h \)[/tex]?
Insufficient information about the domain of [tex]\( h \)[/tex] is provided.
Answer: Information about the domain of [tex]\( h(x) \)[/tex] is not provided.
##### 13) Is the number 6 a member of the range of function [tex]\( h \)[/tex]?
Insufficient information about the range of [tex]\( h \)[/tex] is provided.
Answer: Information about the range of [tex]\( h(x) \)[/tex] is not provided.
##### 14) Is the number -5.5 a member of the range of function [tex]\( h \)[/tex]?
Insufficient information about the range of [tex]\( h \)[/tex] is provided.
Answer: Information about the range of [tex]\( h(x) \)[/tex] is not provided.
It's important to note that the lack of information about [tex]\( h \)[/tex] makes it impossible to accurately answer several questions. If additional information about [tex]\( h \)[/tex] is provided, these questions could be revisited.
