To find the correct decompositions of the function [tex]\( h(x) = \sqrt{7x + 2} - 10 \)[/tex] into [tex]\( h(x) = f(g(x)) \)[/tex], we need to check each given pair of functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] and verify if they correctly represent [tex]\( h(x) \)[/tex].
Consider the pairs one by one:
1. [tex]\( f(x) = \sqrt{7} \sqrt{x} - 10 \)[/tex] and [tex]\( g(x) = x + 2 \)[/tex]:
[tex]\[
f(g(x)) = \sqrt{7} \sqrt{x+2} - 10
\][/tex]
However, this does not simplify to [tex]\( h(x) = \sqrt{7x + 2} - 10 \)[/tex]. This pair is incorrect.
2. [tex]\( f(x) = x - 10 \)[/tex] and [tex]\( g(x) = \sqrt{7x + 2} \)[/tex]:
[tex]\[
f(g(x)) = g(x) - 10 = \sqrt{7x + 2} - 10
\][/tex]
This is exactly [tex]\( h(x) = \sqrt{7x + 2} - 10 \)[/tex]. This pair is correct.
3. [tex]\( f(x) = 7x \)[/tex] and [tex]\( g(x) = \sqrt{x + 2} - 10 \)[/tex]:
[tex]\[
f(g(x)) = 7 (\sqrt{x + 2} - 10) = 7 \sqrt{x + 2} - 70
\][/tex]
This does not simplify to [tex]\( h(x) = \sqrt{7x + 2} - 10 \)[/tex]. This pair is incorrect.
4. [tex]\( f(x) = x + 2 \)[/tex] and [tex]\( g(x) = \sqrt{7} \sqrt{x} - 10 \)[/tex]:
[tex]\[
f(g(x)) = g(x) + 2 = \sqrt{7} \sqrt{x} - 10 + 2 = \sqrt{7} \sqrt{x} - 8
\][/tex]
This does not simplify to [tex]\( h(x) = \sqrt{7x + 2} - 10 \)[/tex]. This pair is incorrect.
5. [tex]\( f(x) = \sqrt{x} - 10 \)[/tex] and [tex]\( g(x) = 7x + 2 \)[/tex]:
[tex]\[
f(g(x)) = \sqrt{g(x)} - 10 = \sqrt{7x + 2} - 10
\][/tex]
This is exactly [tex]\( h(x) = \sqrt{7x + 2} - 10 \)[/tex]. This pair is correct.
Therefore, the correct answers are:
- [tex]\( f(x) = x - 10 \)[/tex] and [tex]\( g(x) = \sqrt{7x + 2} \)[/tex] (Choice 2)
- [tex]\( f(x) = \sqrt{x} - 10 \)[/tex] and [tex]\( g(x) = 7x + 2 \)[/tex] (Choice 5)
The correct decompositions are given in choices 2 and 5.
