To determine the correct function [tex]\( f(x) \)[/tex] from the given choices such that [tex]\( h(x) = f(g(x)) \)[/tex], let's proceed step-by-step.
We are given:
1. [tex]\( h(x) = \frac{1}{x^2 + 1} \)[/tex]
2. [tex]\( g(x) = x^2 + 1 \)[/tex]
3. [tex]\( h(x) = f(g(x)) \)[/tex]
We need to find [tex]\( f \)[/tex] in terms of [tex]\( x \)[/tex] that satisfies the relationship when composed with [tex]\( g(x) \)[/tex].
First, express [tex]\( h(x) \)[/tex] using the function [tex]\( f(x) \)[/tex] and the given function [tex]\( g(x) \)[/tex].
So, we have:
[tex]\[ h(x) = f(g(x)) \][/tex]
Since [tex]\( g(x) = x^2 + 1 \)[/tex], substitute [tex]\( g(x) \)[/tex] into [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = f(g(x)) = f(x^2 + 1) \][/tex]
But we know that:
[tex]\[ h(x) = \frac{1}{x^2 + 1} \][/tex]
Therefore, we want:
[tex]\[ f(x^2 + 1) = \frac{1}{x^2 + 1} \][/tex]
Now, we need to find [tex]\( f \)[/tex] such that [tex]\( f(u) = \frac{1}{u} \)[/tex].
Let's check each of the given choices to see which one matches this condition.
### Checking the Choices:
1. [tex]\( f(x) = \frac{1}{\sqrt{x}} \)[/tex]:
[tex]\[ f(x^2 + 1) = \frac{1}{\sqrt{x^2 + 1}} \][/tex]
This is not equal to [tex]\( \frac{1}{x^2 + 1} \)[/tex], so this choice is incorrect.
2. [tex]\( f(x) = \frac{1}{x} \)[/tex]:
[tex]\[ f(x^2 + 1) = \frac{1}{x^2 + 1} \][/tex]
This matches [tex]\( \frac{1}{x^2 + 1} \)[/tex]. So this choice is correct.
3. [tex]\( f(x) = \frac{1}{x+1} \)[/tex]:
[tex]\[ f(x^2 + 1) = \frac{1}{(x^2 + 1) + 1} = \frac{1}{x^2 + 2} \][/tex]
This does not match [tex]\( \frac{1}{x^2 + 1} \)[/tex], so this choice is incorrect.
4. [tex]\( f(x) = \frac{1}{x^2 + 1} \)[/tex]:
[tex]\[ f(x^2 + 1) = \frac{1}{(x^2 + 1)^2 + 1} \][/tex]
This does not match [tex]\( \frac{1}{x^2 + 1} \)[/tex], so this choice is incorrect.
Thus, the correct function [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = \frac{1}{x} \][/tex]
