Answer :
To evaluate the function [tex]\( f(x) = x^2 + 2x + 9 \)[/tex] at the given values, follow these steps:
### Part a: Evaluate [tex]\( f(8) \)[/tex]
1. Substitute [tex]\( x = 8 \)[/tex] into the function:
Plug [tex]\( 8 \)[/tex] into [tex]\( f(x) = x^2 + 2x + 9 \)[/tex].
[tex]\[
f(8) = 8^2 + 2 \times 8 + 9
\][/tex]
2. Calculate each term:
- [tex]\( 8^2 = 64 \)[/tex]
- [tex]\( 2 \times 8 = 16 \)[/tex]
3. Add the values together:
[tex]\[
f(8) = 64 + 16 + 9 = 89
\][/tex]
So, the simplified value is [tex]\( f(8) = 89 \)[/tex].
### Part b: Evaluate [tex]\( f(x+2) \)[/tex]
1. Substitute [tex]\( x = x+2 \)[/tex] into the function:
Replace [tex]\( x \)[/tex] with [tex]\( (x+2) \)[/tex] in [tex]\( f(x) = x^2 + 2x + 9 \)[/tex].
[tex]\[
f(x+2) = (x+2)^2 + 2(x+2) + 9
\][/tex]
2. Expand each term:
- [tex]\( (x+2)^2 = x^2 + 4x + 4 \)[/tex]
- [tex]\( 2(x+2) = 2x + 4 \)[/tex]
3. Combine all terms:
[tex]\[
f(x+2) = x^2 + 4x + 4 + 2x + 4 + 9
\][/tex]
4. Simplify by combining like terms:
- Combine the [tex]\( x \)[/tex] terms: [tex]\( 4x + 2x = 6x \)[/tex]
- Combine the constant terms: [tex]\( 4 + 4 + 9 = 17 \)[/tex]
So, [tex]\( f(x+2) = x^2 + 6x + 17 \)[/tex].
Therefore, the simplified expression for [tex]\( f(x+2) \)[/tex] is [tex]\( x^2 + 6x + 17 \)[/tex].
### Part a: Evaluate [tex]\( f(8) \)[/tex]
1. Substitute [tex]\( x = 8 \)[/tex] into the function:
Plug [tex]\( 8 \)[/tex] into [tex]\( f(x) = x^2 + 2x + 9 \)[/tex].
[tex]\[
f(8) = 8^2 + 2 \times 8 + 9
\][/tex]
2. Calculate each term:
- [tex]\( 8^2 = 64 \)[/tex]
- [tex]\( 2 \times 8 = 16 \)[/tex]
3. Add the values together:
[tex]\[
f(8) = 64 + 16 + 9 = 89
\][/tex]
So, the simplified value is [tex]\( f(8) = 89 \)[/tex].
### Part b: Evaluate [tex]\( f(x+2) \)[/tex]
1. Substitute [tex]\( x = x+2 \)[/tex] into the function:
Replace [tex]\( x \)[/tex] with [tex]\( (x+2) \)[/tex] in [tex]\( f(x) = x^2 + 2x + 9 \)[/tex].
[tex]\[
f(x+2) = (x+2)^2 + 2(x+2) + 9
\][/tex]
2. Expand each term:
- [tex]\( (x+2)^2 = x^2 + 4x + 4 \)[/tex]
- [tex]\( 2(x+2) = 2x + 4 \)[/tex]
3. Combine all terms:
[tex]\[
f(x+2) = x^2 + 4x + 4 + 2x + 4 + 9
\][/tex]
4. Simplify by combining like terms:
- Combine the [tex]\( x \)[/tex] terms: [tex]\( 4x + 2x = 6x \)[/tex]
- Combine the constant terms: [tex]\( 4 + 4 + 9 = 17 \)[/tex]
So, [tex]\( f(x+2) = x^2 + 6x + 17 \)[/tex].
Therefore, the simplified expression for [tex]\( f(x+2) \)[/tex] is [tex]\( x^2 + 6x + 17 \)[/tex].
