To express the function [tex]\( y = \sqrt{x^{19} + 9} \)[/tex] as a composition of two simpler functions, we need to find functions [tex]\( u \)[/tex] and [tex]\( y \)[/tex] such that [tex]\( y \)[/tex] is expressed in terms of [tex]\( u \)[/tex]. Essentially, we want to split the original function into two functions where one depends on [tex]\( x \)[/tex] and the second depends on the first function.
Given the original function:
[tex]\[ y = \sqrt{x^{19} + 9} \][/tex]
We will check each pair of simpler functions to see if they correctly decompose this function.
1. [tex]\( u = x^{19} + 9 \)[/tex] and [tex]\( y = \sqrt{u} \)[/tex]
[tex]\[
u = x^{19} + 9 \quad \Rightarrow \quad y = \sqrt{u} = \sqrt{x^{19} + 9}
\][/tex]
This pair is valid because substituting [tex]\( u \)[/tex] into [tex]\( y \)[/tex] results in the original function [tex]\( y = \sqrt{x^{19} + 9} \)[/tex].
2. [tex]\( u = x + 9 \)[/tex] and [tex]\( y = \sqrt{u^{19}} \)[/tex]
[tex]\[
u = x + 9 \quad \Rightarrow \quad y = \sqrt{u^{19}} = \sqrt{(x+9)^{19}}
\][/tex]
This pair does not match the original function because [tex]\( \sqrt{(x+9)^{19}} \neq \sqrt{x^{19} + 9} \)[/tex].
3. [tex]\( u = \sqrt{x} \)[/tex] and [tex]\( y = u^{19} + 9 \)[/tex]
[tex]\[
u = \sqrt{x} \quad \Rightarrow \quad y = u^{19} + 9 = (\sqrt{x})^{19} + 9 = x^{9.5} + 9
\][/tex]
This pair does not match the original function because [tex]\( x^{9.5} + 9 \neq \sqrt{x^{19} + 9} \)[/tex].
4. [tex]\( u = x^{19} \)[/tex] and [tex]\( y = \sqrt{u + 9} \)[/tex]
[tex]\[
u = x^{19} \quad \Rightarrow \quad y = \sqrt{u + 9} = \sqrt{x^{19} + 9}
\][/tex]
This pair is also valid because substituting [tex]\( u \)[/tex] into [tex]\( y \)[/tex] results in the original function [tex]\( y = \sqrt{x^{19} + 9} \)[/tex].
5. [tex]\( u = \sqrt{x + 9} \)[/tex] and [tex]\( y = u^{19} \)[/tex]
[tex]\[
u = \sqrt{x + 9} \quad \Rightarrow \quad y = u^{19} = (\sqrt{x + 9})^{19} = (x + 9)^{9.5}
\][/tex]
This pair does not match the original function because [tex]\( (x + 9)^{9.5} \neq \sqrt{x^{19} + 9} \)[/tex].
Thus, the pairs of functions that correctly express [tex]\( y = \sqrt{x^{19} + 9} \)[/tex] are:
[tex]\[ u = x^{19} + 9 \quad \text{and} \quad y = \sqrt{u} \][/tex]
[tex]\[ u = x^{19} \quad \text{and} \quad y = \sqrt{u + 9} \][/tex]
These pairs decompose the original function as desired.
