Answer :
Let's evaluate the function [tex]\( f(x) = x^2 + 2x - 7 \)[/tex] at the given values step by step.
### a. Evaluate [tex]\( f(-5) \)[/tex]
We start by substituting [tex]\( x = -5 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-5) = (-5)^2 + 2(-5) - 7 \][/tex]
First, calculate [tex]\((-5)^2\)[/tex]:
[tex]\[ (-5)^2 = 25 \][/tex]
Next, calculate [tex]\( 2(-5) \)[/tex]:
[tex]\[ 2(-5) = -10 \][/tex]
Finally, combine the results:
[tex]\[ f(-5) = 25 - 10 - 7 = 15 - 7 = 8 \][/tex]
So, [tex]\( f(-5) = 8 \)[/tex].
### b. Evaluate [tex]\( f(x + 4) \)[/tex]
Now we substitute [tex]\( x + 4 \)[/tex] in place of [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x + 4) = (x + 4)^2 + 2(x + 4) - 7 \][/tex]
Expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[ (x + 4)^2 = x^2 + 8x + 16 \][/tex]
Expand [tex]\( 2(x + 4) \)[/tex]:
[tex]\[ 2(x + 4) = 2x + 8 \][/tex]
Now, combine all terms:
[tex]\[ f(x + 4) = x^2 + 8x + 16 + 2x + 8 - 7 \][/tex]
Simplify this expression:
[tex]\[ f(x + 4) = x^2 + (8x + 2x) + (16 + 8 - 7) \][/tex]
[tex]\[ f(x + 4) = x^2 + 10x + 17 \][/tex]
So [tex]\( f(x + 4) = x^2 + 10x + 17 \)[/tex].
### c. Evaluate [tex]\( f(-x) \)[/tex]
Finally, substitute [tex]\(-x\)[/tex] in place of [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-x) = (-x)^2 + 2(-x) - 7 \][/tex]
First, [tex]\((-x)^2\)[/tex]:
[tex]\[ (-x)^2 = x^2 \][/tex]
Next, [tex]\( 2(-x) \)[/tex]:
[tex]\[ 2(-x) = -2x \][/tex]
Combine all terms:
[tex]\[ f(-x) = x^2 - 2x - 7 \][/tex]
So [tex]\( f(-x) = x^2 - 2x - 7 \)[/tex].
### Summary
- [tex]\( f(-5) = 8 \)[/tex]
- [tex]\( f(x+4) = x^2 + 10x + 17 \)[/tex]
- [tex]\( f(-x) = x^2 - 2x - 7 \)[/tex]
Therefore, the simplified results are:
- [tex]\( f(-5) = 8 \)[/tex]
- [tex]\( f(x+4) = x^2 + 10x + 17 \)[/tex]
- [tex]\( f(-x) = x^2 - 2x - 7 \)[/tex]
### a. Evaluate [tex]\( f(-5) \)[/tex]
We start by substituting [tex]\( x = -5 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-5) = (-5)^2 + 2(-5) - 7 \][/tex]
First, calculate [tex]\((-5)^2\)[/tex]:
[tex]\[ (-5)^2 = 25 \][/tex]
Next, calculate [tex]\( 2(-5) \)[/tex]:
[tex]\[ 2(-5) = -10 \][/tex]
Finally, combine the results:
[tex]\[ f(-5) = 25 - 10 - 7 = 15 - 7 = 8 \][/tex]
So, [tex]\( f(-5) = 8 \)[/tex].
### b. Evaluate [tex]\( f(x + 4) \)[/tex]
Now we substitute [tex]\( x + 4 \)[/tex] in place of [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x + 4) = (x + 4)^2 + 2(x + 4) - 7 \][/tex]
Expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[ (x + 4)^2 = x^2 + 8x + 16 \][/tex]
Expand [tex]\( 2(x + 4) \)[/tex]:
[tex]\[ 2(x + 4) = 2x + 8 \][/tex]
Now, combine all terms:
[tex]\[ f(x + 4) = x^2 + 8x + 16 + 2x + 8 - 7 \][/tex]
Simplify this expression:
[tex]\[ f(x + 4) = x^2 + (8x + 2x) + (16 + 8 - 7) \][/tex]
[tex]\[ f(x + 4) = x^2 + 10x + 17 \][/tex]
So [tex]\( f(x + 4) = x^2 + 10x + 17 \)[/tex].
### c. Evaluate [tex]\( f(-x) \)[/tex]
Finally, substitute [tex]\(-x\)[/tex] in place of [tex]\( x \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-x) = (-x)^2 + 2(-x) - 7 \][/tex]
First, [tex]\((-x)^2\)[/tex]:
[tex]\[ (-x)^2 = x^2 \][/tex]
Next, [tex]\( 2(-x) \)[/tex]:
[tex]\[ 2(-x) = -2x \][/tex]
Combine all terms:
[tex]\[ f(-x) = x^2 - 2x - 7 \][/tex]
So [tex]\( f(-x) = x^2 - 2x - 7 \)[/tex].
### Summary
- [tex]\( f(-5) = 8 \)[/tex]
- [tex]\( f(x+4) = x^2 + 10x + 17 \)[/tex]
- [tex]\( f(-x) = x^2 - 2x - 7 \)[/tex]
Therefore, the simplified results are:
- [tex]\( f(-5) = 8 \)[/tex]
- [tex]\( f(x+4) = x^2 + 10x + 17 \)[/tex]
- [tex]\( f(-x) = x^2 - 2x - 7 \)[/tex]
