Question

the following three functions look very similar, but define very different functions. think about how they are defined and write each as a composition of functions given the information below. (none of the functions in your compositions should be the same as the given function.) 1. $cos^4(x) = f(g(x))$ where $f(x) = square$, and $g(x) = square$. 2. $cos(cos(x)) = f(g(x))$ where $f(x) = square$, and $g(x) = square$. 3. $cos x^4 = f(g(x))$ where $f(x) = square$, and $g(x) = square$.

Explanation:

Problem 1: $\boldsymbol{\cos^4(x) = f(g(x))}$

Step1: Identify the inner and outer functions

The function $\cos^4(x)$ can be seen as taking $\cos(x)$ and then raising it to the 4th power. So the inner function $g(x)$ should be $\cos(x)$, and the outer function $f(x)$ should be a function that raises its input to the 4th power.

Step2: Define $f(x)$ and $g(x)$

Let $g(x)=\cos(x)$ (this is the inner function, taking the input $x$ and applying the cosine function). Then $f(x)=x^4$ (this is the outer function, taking the result of $g(x)$ and raising it to the 4th power). When we compose $f$ and $g$, we get $f(g(x)) = f(\cos(x)) = (\cos(x))^4=\cos^4(x)$, which matches the given function.

Step1: Identify the inner and outer functions

The function $\cos(\cos(x))$ is composed of taking $\cos(x)$ first (inner function) and then applying the cosine function again (outer function).

Step2: Define $f(x)$ and $g(x)$

Let $g(x)=\cos(x)$ (the inner function, applying cosine to $x$). Then $f(x)=\cos(x)$ (the outer function, applying cosine to the result of $g(x)$). Composing them, $f(g(x)) = f(\cos(x))=\cos(\cos(x))$, which is the given function.

Step1: Identify the inner and outer functions

The function $\cos(x^4)$ can be seen as taking $x^4$ (inner function) and then applying the cosine function (outer function) to that result.

Step2: Define $f(x)$ and $g(x)$

Let $g(x)=x^4$ (the inner function, raising $x$ to the 4th power). Then $f(x)=\cos(x)$ (the outer function, applying cosine to the result of $g(x)$). Composing them, $f(g(x)) = f(x^4)=\cos(x^4)$, which matches the given function.

Answer:

(for Problem 1):
$f(x) = x^4$, $g(x) = \cos(x)$

Problem 2: $\boldsymbol{\cos(\cos(x)) = f(g(x))}$