Question

Let f(x)=9−8x‾‾‾‾‾‾√ .Which of the following decompositions of f(x)=p(q(x)) into a pair of functions p(x) (the outside function) and q(x) (the inside function) is/are correct ? Select all that apply if more than one is appropriate. A. p(x)=−6x‾‾‾‾√ and q(x)=9+x

B. p(x)=9+x‾‾‾‾‾√ and q(x)=6x
C. p(x)=−x‾‾‾√ and q(x)=6x−9
D. p(x)=9−x‾‾‾‾‾√ and q(x)=6x
E. p(x)=x√ and q(x)=9−6x
F. p(x)=9−6x and q(x)=x√
G. All of the above
H. None of the above

Answer 1

After analyzing each option, C (p(x)=−x and q(x)=6x−9) and E (p(x)=x√ and q(x)=9−6x) are correct decompositions of the function f(x)=9−8x, as they both accurately reflect the composition of f(x) when p(x) is applied to q(x).

Step-by-step explanation:

The student is asking for a decomposition of the function f(x)=9−8x as a composition of two functions p(x) and q(x). To find the correct decompositions, we need to see which combinations of p(x) applied to q(x) will result in the original function f(x). For a function f(x) that is expressed as p(q(x)), the inside function q(x) is calculated first, and then the outside function p(x) is applied to the result of q(x).

Examining each option individually:

• Option A: The composition p(q(x)) results in p(9+x) = -6 − 6(9+x), which does not equal f(x).
• Option B cannot work as q(x) = 6x makes no sense in the original function.
• Option C: p(q(x)) is p(6x-9) = −(6x-9) which does equal f(x) after taking the square root of the result.
• Option D: Does not make sense as q(x) is again 6x.
• Option E: p(q(x)) is −(9-6x), which equals f(x).
• Option F: This cannot be true since p(x) is not involving a root at all and q(x) just a simple square root.

Considering the correct operations, options C and E are valid decompositions of f(x) into functions p(x) and q(x).

Answer 2

C, D and E

Step-by-step explanation:

Assuming

f(x) = √(9−8x)

and the options are:

A. p(x) = √(-8x) and q(x)=9+x

B. p(x) = √(9+x) and q(x)=8x

C. p(x) = √(-x) and q(x)=8x−9

D. p(x)=√(9−x) and q(x)=8x

E. p(x)=√x and q(x)=9−8x

F. p(x)=9−8x and q(x)=√x

G. All of the above

H. None of the above

Substituting in f(x)=p(q(x)) we get:

A. √(-8(9+x)) = √(-72 - 8x)

B. √(9+(8x)) = √(9+8x)

C. √(-(8x−9 )) = √(-8x + 9)

D. √(9−(8x)) = √(9−8x)

E. √(9−8x )

F. 9−8(√x ) = 9 - 8√x