A composite function can be written as $$w(u(x))$$, where $${u}$$ and $${w}$$ are basic functions. Is $${g}({x})=\sqrt{{e}^{{x}}}$$ a composite function? If so, what are the "inner" and "outer" functions?



<div id="displayOption0" class="markdown-body editormd-html-preview"><textarea>g is composite, inner function $$=e^{x}$$, outer function= $$=\sqrt{x}$$

</textarea>g is composite, inner function <math><semantics><mrow><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">=e^{x}</annotation></semantics></math>=e​x​​, outer function= <math><semantics><mrow><mo>=</mo><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">=\sqrt{x}</annotation></semantics></math>=√​x​​​
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<div id="displayOption1" class="markdown-body editormd-html-preview"><textarea>g is composite, inner function $$=\sqrt{x}$$, outer function=$$e^{x}$$

</textarea>g is composite, inner function <math><semantics><mrow><mo>=</mo><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">=\sqrt{x}</annotation></semantics></math>=√​x​​​, outer function=<math><semantics><mrow><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">e^{x}</annotation></semantics></math>e​x​​
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<div id="displayOption2" class="markdown-body editormd-html-preview"><textarea>$$g$$ is not a composite function.

</textarea><math><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>g is not a composite function.
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<div id="displayOption3" class="markdown-body editormd-html-preview"><textarea>None of the above
</textarea>None of the above
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<div id="id_cont_display" class="markdown-body editormd-html-preview"><textarea>A composite function can be written as $$w(u(x))$$, where $${u}$$ and $${w}$$ are basic functions. Is $${g}({x})=\sqrt{{e}^{{x}}}$$ a composite function? If so, what are the "inner" and "outer" functions?

</textarea>A composite function can be written as <math><semantics><mrow><mi>w</mi><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">w(u(x))</annotation></semantics></math>w(u(x)), where <math><semantics><mrow><mrow><mi>u</mi></mrow></mrow><annotation encoding="application/x-tex">{u}</annotation></semantics></math>u and <math><semantics><mrow><mrow><mi>w</mi></mrow></mrow><annotation encoding="application/x-tex">{w}</annotation></semantics></math>w are basic functions. Is <math><semantics><mrow><mrow><mi>g</mi></mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo><mo>=</mo><msqrt><mrow><msup><mrow><mi>e</mi></mrow><mrow><mrow><mi>x</mi></mrow></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">{g}({x})=\sqrt{{e}^{{x}}}</annotation></semantics></math>g(x)=√​e​x​​​​​ a composite function? If so, what are the “inner” and “outer” functions?
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>$$g(x)$$ can be Dritten as $$\omega(u(x))$$ where,

$$u(x)=e^{x}, \quad \omega(x)=\sqrt{x}$$

So, $$g(x)$$ is composite function,

Inner function $$=e^{x}$$

Outer function $$=\sqrt{x}$$


</textarea><math><semantics><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math>g(x) can be Dritten as <math><semantics><mrow><mi>ω</mi><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><annotation encoding="application/x-tex">\omega(u(x))</annotation></semantics></math>ω(u(x)) where,
<math><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>ω</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">u(x)=e^{x}, \quad \omega(x)=\sqrt{x}</annotation></semantics></math>u(x)=e​x​​,ω(x)=√​x​​​
So, <math><semantics><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">g(x)</annotation></semantics></math>g(x) is composite function,
Inner function <math><semantics><mrow><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">=e^{x}</annotation></semantics></math>=e​x​​
Outer function <math><semantics><mrow><mo>=</mo><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">=\sqrt{x}</annotation></semantics></math>=√​x​​​
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