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

A. [tex]h^2 + 9h + 23[/tex]
B. [tex](3+h)^2 + 8 + h[/tex]
C. [tex]h^2 + h + 23[/tex]
D. [tex]\left(x^2 + 3x + 5\right)(3+h)[/tex]

Sagot :

GinnyAnswer
Sure! Let's solve the given problem step-by-step.

We are given the function [tex]\( f(x) = x^2 + 3x + 5 \)[/tex] and we are asked to find [tex]\( f(3+h) \)[/tex].

1. Substitute [tex]\( x \)[/tex] with [tex]\( 3+h \)[/tex] in the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(3+h) = (3+h)^2 + 3(3+h) + 5 \][/tex]

2. Expand the squared term [tex]\((3+h)^2\)[/tex]:
[tex]\[ (3+h)^2 = 3^2 + 2 \cdot 3 \cdot h + h^2 = 9 + 6h + h^2 \][/tex]

3. Distribute and simplify the other term [tex]\( 3(3+h) \)[/tex]:
[tex]\[ 3(3+h) = 3 \cdot 3 + 3 \cdot h = 9 + 3h \][/tex]

4. Substitute back and combine all terms:
[tex]\[ f(3+h) = 9 + 6h + h^2 + 9 + 3h + 5 \][/tex]

5. Add up all like terms:
[tex]\[ h^2 + (6h + 3h) + (9 + 9 + 5) = h^2 + 9h + 23 \][/tex]

Thus, the function [tex]\( f(3+h) \)[/tex] simplifies to:
[tex]\[ f(3+h) = h^2 + 9h + 23 \][/tex]

As per the choices given, the correct answer is:
A. [tex]\( h^2 + 9h + 23 \)[/tex]
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