A composite function can be written as $$w(u(x))$$ where $${u}$$ and $${w}$$ are basic functions. Is $${h}({x})={e}^{-x^{2}}$$ a composite function? If so, what are $$u$$ and $$w$$ ?



<div id="displayOption0" class="markdown-body editormd-html-preview"><textarea>h is composite, where $$u(x)=-x^{2}, w(x)=e^{x}$$

</textarea>h is composite, where <math><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo separator="true">,</mo><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">u(x)=-x^{2}, w(x)=e^{x}</annotation></semantics></math>u(x)=−x​2​​,w(x)=e​x​​
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<div id="displayOption1" class="markdown-body editormd-html-preview"><textarea>h is composite where $$u(x)=e^{x}, w(x)=-x^{2}$$

</textarea>h is composite 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><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">u(x)=e^{x}, w(x)=-x^{2}</annotation></semantics></math>u(x)=e​x​​,w(x)=−x​2​​
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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 $${h}({x})={e}^{-x^{2}}$$ a composite function? If so, what are $$u$$ and $$w$$ ?

</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>e</mi></mrow><mrow><mo>−</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex">{h}({x})={e}^{-x^{2}}</annotation></semantics></math>h(x)=e​−x​2​​​​ a composite function? If so, what are <math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>u and <math><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>w ?
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>$$h(x)$$ can be written as, $$h(x)=e^{f(x)}$$ where $$f(x)=-x^{2}$$ So output of $$f(x)$$ is the input of $$h(x)$$.

So, $$h$$ is a composite function

so, $$u(x)=-x^{2}, w(x)=e^{x}$$
</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>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">h(x)=e^{f(x)}</annotation></semantics></math>h(x)=e​f(x)​​ where <math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f(x)=-x^{2}</annotation></semantics></math>f(x)=−x​2​​ So output of <math><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math>f(x) is the input of <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).
So, <math><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>h is a composite function
so, <math><semantics><mrow><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo separator="true">,</mo><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">u(x)=-x^{2}, w(x)=e^{x}</annotation></semantics></math>u(x)=−x​2​​,w(x)=e​x​​
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