Function Composition: Understanding \(f(f(f(x)))\)

This problem asks us to evaluate a complex function expression involving nested functions. We are given the function \(f(x) = x^2\) and need to find the value of \(\frac{{f(f\left( {f\left( x \right))} \right)}}{{f\left( x \right)}}\).

Function Definition

The core of this problem lies in understanding the given function:

• The function is defined as \(f(x) = x^2\). This means that for any input, the function outputs the square of that input.
• The domain for \(x\) is \((-\infty, \infty)\), which covers all real numbers.

Nested Functions Evaluation: Step-by-Step

Let's evaluate the nested function \(f(f(f(x)))\) step-by-step from the innermost part outwards.

First Level: \(f(x)\)

The innermost function is simply \(f(x)\).

• Given: \(f(x) = x^2\)

Second Level: \(f(f(x))\)

Now we evaluate \(f\) of the result from the first level. Replace the input \(x\) in \(f(x)\) with \(f(x)\) itself.

• We have \(f(x) = x^2\).
• So, \(f(f(x)) = f(x^2)\).
• Applying the function definition \(f(\text{input}) = (\text{input})^2\) to \(x^2\) as the input: \(f(x^2) = (x^2)^2\)
• Using the exponent rule \((a^m)^n = a^{mn}\): \((x^2)^2 = x^{2 \times 2} = x^4\)
• Therefore, \(f(f(x)) = x^4\).

Third Level: \(f(f(f(x)))\)

Finally, we evaluate \(f\) of the result from the second level. Replace the input \(x\) in \(f(x)\) with \(f(f(x))\), which we found to be \(x^4\).

• We have \(f(f(x)) = x^4\).
• So, \(f(f(f(x))) = f(x^4)\).
• Applying the function definition \(f(\text{input}) = (\text{input})^2\) to \(x^4\) as the input: \(f(x^4) = (x^4)^2\)
• Using the exponent rule \((a^m)^n = a^{mn}\) again: \((x^4)^2 = x^{4 \times 2} = x^8\)
• Therefore, \(f(f(f(x))) = x^8\).

Evaluating the Overall Expression

Now that we have both the numerator \(f(f(f(x))) = x^8\) and the denominator \(f(x) = x^2\), we can substitute these into the given expression:

\(\frac{{f(f\left( {f\left( x \right))} \right)}}{{f\left( x \right)}} = \frac{x^8}{x^2}\)

Using the exponent rule for division \(\frac{a^m}{a^n} = a^{m-n}\):

\(\frac{x^8}{x^2} = x^{8-2} = x^6\)

So, the expression simplifies to \(x^6\).

Expressing the Result in Terms of \(f(x)\)

The options are given in terms of \(f(x)\), so we need to express \(x^6\) using \(f(x)\). Recall that \(f(x) = x^2\).

We can rewrite \(x^6\) as:

\(x^6 = (x^2)^3\)

Since \(f(x) = x^2\), we can substitute \(f(x)\) for \(x^2\):

\((x^2)^3 = (f(x))^3\)

Therefore, \(\frac{{f(f\left( {f\left( x \right))} \right)}}{{f\left( x \right)}} = (f(x))^3\).

Summary of Calculation Steps

The final result matches option 3.