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If [tex]f(x) = x^2 + 3x + 5[/tex], what is [tex]f(a+h)[/tex]?

A. [tex]a^2 + 2ah + h^2 + 3a + 3h + 5[/tex]

B. [tex]h^2 + 3a + 3h + 5[/tex]

C. [tex](x^2 + 3ax + 5)(a+h)[/tex]

D. [tex](a+h)^2 + 3(a+h) + 5[/tex]

Sagot :

GinnyAnswer
To solve for [tex]\( f(a+h) \)[/tex] where the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex], we need to substitute [tex]\( a + h \)[/tex] in place of [tex]\( x \)[/tex] in the given function and then expand and simplify the expression.

Given:
[tex]\[ f(x) = x^2 + 3x + 5 \][/tex]

1. Substitute [tex]\( a + h \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[ f(a + h) = (a + h)^2 + 3(a + h) + 5 \][/tex]

2. Expand [tex]\( (a + h)^2 \)[/tex]:
[tex]\[ (a + h)^2 = a^2 + 2ah + h^2 \][/tex]

3. Expand [tex]\( 3(a + h) \)[/tex]:
[tex]\[ 3(a + h) = 3a + 3h \][/tex]

4. Substitute the expanded forms back into the function:
[tex]\[ f(a + h) = a^2 + 2ah + h^2 + 3a + 3h + 5 \][/tex]

So the expanded and simplified expression for [tex]\( f(a + h) \)[/tex] is:
[tex]\[ a^2 + 2ah + h^2 + 3a + 3h + 5 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{a^2 + 2ah + h^2 + 3a + 3h + 5} \][/tex]

This matches option A.
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