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Evaluate the following functions. Write your answer and complete solution on separate paper.

1. Given [tex]w(n) = n - 1[/tex], find the value of the function if [tex]n = -1[/tex].

2. Given [tex]f(x) = |x| - 3[/tex], find [tex]f(9.3)[/tex].

3. Evaluate the function [tex]w(x) = |-2x + 3|[/tex] if [tex]x = -1[/tex].

4. Evaluate [tex]f(x) = -x - 1[/tex], find [tex]f\left(a^2\right)[/tex].

5. Given [tex]f(x) = 4x - 5[/tex], find [tex]f(2x + 3)[/tex].



Answer :

GinnyAnswer
Sure, let's go through each problem step-by-step.

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### 1. Given [tex]\( w(n) = n - 1 \)[/tex]
Find the value of the function if [tex]\( n = -1 \)[/tex].

Solution:
[tex]\[ w(n) = n - 1 \][/tex]
Substitute [tex]\( n = -1 \)[/tex]:
[tex]\[ w(-1) = -1 - 1 = -2 \][/tex]

Answer: [tex]\( w(-1) = -2 \)[/tex]

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### 2. Given [tex]\( f(x) = |x| - 3 \)[/tex]
Find [tex]\( f(9.3) \)[/tex].

Solution:
[tex]\[ f(x) = |x| - 3 \][/tex]
Substitute [tex]\( x = 9.3 \)[/tex]:
[tex]\[ f(9.3) = |9.3| - 3 \][/tex]
Since [tex]\(|9.3|\)[/tex] is just [tex]\(9.3\)[/tex]:
[tex]\[ f(9.3) = 9.3 - 3 = 6.3 \][/tex]

Answer: [tex]\( f(9.3) = 6.3 \)[/tex]

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### 3. Evaluate the function [tex]\( w(x) = |-2x + 3| \)[/tex]
If [tex]\( x = -1 \)[/tex].

Solution:
[tex]\[ w(x) = |-2x + 3| \][/tex]
Substitute [tex]\( x = -1 \)[/tex]:
[tex]\[ w(-1) = |-2(-1) + 3| \][/tex]
Calculate inside the absolute value:
[tex]\[ w(-1) = |2 + 3| = |5| = 5 \][/tex]

Answer: [tex]\( w(-1) = 5 \)[/tex]

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### 4. Evaluate [tex]\( f(x) = -x - 1 \)[/tex]
Find [tex]\( f(a^2) \)[/tex].

Solution:
[tex]\[ f(x) = -x - 1 \][/tex]
Substitute [tex]\( x = a^2 \)[/tex]:
[tex]\[ f(a^2) = -(a^2) - 1 \][/tex]

Use [tex]\( a = 1 \)[/tex]
[tex]\[ f(1^2) = -1 - 1 = -2 \][/tex]

Answer: [tex]\( f(a^2) = -2 \)[/tex]

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### 5. Given [tex]\( f(x) = 4x - 5 \)[/tex]
Find [tex]\( f(2x + 3) \)[/tex].

Solution:
[tex]\[ f(x) = 4x - 5 \][/tex]
Substitute [tex]\( x = 2x + 3 \)[/tex]:
[tex]\[ f(2x + 3) = 4(2x + 3) - 5 \][/tex]
Distribute the 4:
[tex]\[ f(2x + 3) = 8x + 12 - 5 \][/tex]
Combine like terms:
[tex]\[ f(2x + 3) = 8x + 7 \][/tex]

Given an example [tex]\( x = 1 \)[/tex]
[tex]\[ f(2(1) + 3) = 8(1) + 7 = 15 \][/tex]

Answer: [tex]\( f(2x + 3) = 15 \)[/tex]

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These are the step-by-step solutions for the given problems.

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