A composite function can be written as $$w(u(x))$$, where $${u}$$ and $${w}$$ are basic functions. Is $${h}({x})={c o s}^{{9}} {x}$$ a composite function? If so, what are the "inner" and "outer" functions?



<div id="displayOption0" class="markdown-body editormd-html-preview"><textarea>$$h$$ is composite. The 'inner ' function is $$\cos (x)$$ and outer function is $$x^{9}$$.

</textarea><math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>h is composite. The ‘inner ‘ function is <math><semantics><mrow><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\cos (x)</annotation></semantics></math>cos(x) and outer function is <math><semantics><mrow><msup><mi>x</mi><mrow><mn>9</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{9}</annotation></semantics></math>x​9​​.
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<div id="displayOption1" class="markdown-body editormd-html-preview"><textarea> $$h$$ is composite The 'inner' function is $$x^{9}$$ and outer function is $$\cos (x)$$.

</textarea> <math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>h is composite The ‘inner’ function is <math><semantics><mrow><msup><mi>x</mi><mrow><mn>9</mn></mrow></msup></mrow><annotation encoding="application/x-tex">x^{9}</annotation></semantics></math>x​9​​ and outer function is <math><semantics><mrow><mi>cos</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\cos (x)</annotation></semantics></math>cos(x).
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<div id="displayOption2" class="markdown-body editormd-html-preview"><textarea>$$h$$ is not a composite function.

</textarea><math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>h 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 $${h}({x})={c o s}^{{9}} {x}$$ 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>h</mi></mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo>)</mo><mo>=</mo><msup><mrow><mi>c</mi><mi>o</mi><mi>s</mi></mrow><mrow><mrow><mn>9</mn></mrow></mrow></msup><mrow><mi>x</mi></mrow></mrow><annotation encoding="application/x-tex">{h}({x})={c o s}^{{9}} {x}</annotation></semantics></math>h(x)=cos​9​​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>$$h(x)$$ can be written as $$w(u(x))$$ where,

$$u(x)=\cos x, \omega(x)=x^{9} \text {, }$$

So, it is a composite function.

Inner function $$=u(x)=\cos x$$

Outer function $$=\omega(x)=x^{9}$$

</textarea><math><semantics><mrow><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">h(x)</annotation></semantics></math>h(x) 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><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>cos</mi><mi>x</mi><mo separator="true">,</mo><mi>ω</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mrow><mn>9</mn></mrow></msup><mtext><mi mathvariant="normal">,</mi><mtext>&nbsp;</mtext></mtext></mrow><annotation encoding="application/x-tex">u(x)=\cos x, \omega(x)=x^{9} \text {, }</annotation></semantics></math>u(x)=cosx,ω(x)=x​9​​,&nbsp;
So, it is a composite function.
Inner function <math><semantics><mrow><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>cos</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">=u(x)=\cos x</annotation></semantics></math>=u(x)=cosx
Outer function <math><semantics><mrow><mo>=</mo><mi>ω</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mrow><mn>9</mn></mrow></msup></mrow><annotation encoding="application/x-tex">=\omega(x)=x^{9}</annotation></semantics></math>=ω(x)=x​9​​
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