IDNLearn.com provides a comprehensive platform for finding accurate answers. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.
Sagot :
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.
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.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
