headleym7
Answered

Given the function [tex]\( h(x) \)[/tex] below, select the answer choice(s) which correctly decompose [tex]\( h(x) \)[/tex] into component functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] so that [tex]\( h(x)=f(g(x)) \)[/tex].

[tex]\[ h(x)=\sqrt{7x+2}-10 \][/tex]

Please select two correct answers.

Select all that apply:

A. [tex]\( h(x)=f(g(x)) \)[/tex], where [tex]\( f(x)=\sqrt{7} \sqrt{x}-10 \)[/tex] and [tex]\( g(x)=x+2 \)[/tex]

B. [tex]\( h(x)=f(g(x)) \)[/tex], where [tex]\( f(x)=x-10 \)[/tex] and [tex]\( g(x)=\sqrt{7x+2} \)[/tex]

C. [tex]\( h(x)=f(g(x)) \)[/tex], where [tex]\( f(x)=7x \)[/tex] and [tex]\( g(x)=\sqrt{x+2}-10 \)[/tex]

D. [tex]\( h(x)=f(g(x)) \)[/tex], where [tex]\( f(x)=x+2 \)[/tex] and [tex]\( g(x)=\sqrt{7} \sqrt{x}-10 \)[/tex]

E. [tex]\( h(x)=f(g(x)) \)[/tex], where [tex]\( f(x)=\sqrt{x}-10 \)[/tex] and [tex]\( g(x)=7x+2 \)[/tex]



Answer :

GinnyAnswer
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.

Other Questions