Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Consider the function \(h\) as defined. Find functions \(f\) and \(g\) such that \((f \circ g)(x)=h(x)\). (There are several possible ways to do this.) $$h(x)=\left(11 x^{2}+12 x\right)^{2}$$

Step-by-Step Solution

Answer

Functions are: \(f(u) = u^2\) and \(g(x) = 11x^2 + 12x\).

Step 1: Understanding the Composition

To express a function as the composition of two functions, \(f \circ g\), where \(f(g(x)) = h(x)\), the goal is to identify the 'inner' function \(g(x)\) and the 'outer' function \(f(x)\). This requires recognizing a structure within the given function that can be split into these two parts.

Step 2: Identify the Inner Function

Look for a core expression within \(h(x)\). Here, the expression \(11x^2 + 12x\) can be isolated and considered a fundamental building block. Thus, let \(g(x) = 11x^2 + 12x\), which captures the critical transformation of \(x\) before any additional manipulation (e.g., squaring).

Step 3: Identify the Outer Function

Once the inner function \(g(x) \) is identified, the remainder of the operation in \(h(x)\) involves squaring the expression \(11x^2 + 12x\). Therefore, define \(f(u) = u^2\), where \(u\) represents the output of the inner function. This means \(f(g(x)) = (11x^2 + 12x)^2\), effectively reconstructing \(h(x)\) as desired.

Step 4: Verify the Composition

Check that the composition of \(f \circ g\) reconstructs \(h(x)\). Substitute \(g(x) = 11x^2 + 12x\) into \(f(u)\) to get \(f(g(x)) = f(11x^2 + 12x) = (11x^2 + 12x)^2\). This matches the given function \(h(x)\), confirming the correctness of \(f(x)\) and \(g(x)\).

Key Concepts

Inner Function

In function composition, the inner function is the one that operates directly on the variable, transforming it in some way before any other operations are applied. Here, identifying the inner function is crucial, as it forms the base alteration from which the rest of the function composition is built.
In our exercise, the function given is \(h(x) = (11x^2 + 12x)^2\). We look for a part of this expression that can be isolated and operated upon first.

• The expression \(11x^2 + 12x\) is a natural candidate because it represents a complete transformation of \(x\) before any further operations take place.
• This expression captures all the initial scaling and addition happening directly to \(x\) before it's squared.

Identifying the inner function lets us separate these initial steps from subsequent operations, simplifying our understanding of the entire function composition. Thus, for this example, we choose \[ g(x) = 11x^2 + 12x \] as our inner function.

Outer Function

The outer function in composition takes the result from the inner function and applies an additional operation to it. It is usually a higher-level operation that finishes the transformation started by the inner function.
When solving for \(f(g(x)) = h(x)\), determining the outer function involves analyzing what remains to be done after the inner function has been applied. For the given function \[ h(x) = (11x^2 + 12x)^2 \]we have already identified \[ g(x) = 11x^2 + 12x \]Thus, the outer function must apply the squaring operation to the result of \(g(x)\).

• The squaring of the entire expression \((11x^2 + 12x)^2\) remains after isolating \(g(x)\).
• This indicates that the outer function is simply squaring its input.

Thus, we define the outer function as \[ f(u) = u^2 \]where \(u\) denotes the outcome of the inner function \(g(x)\). This precisely reconstructs \(h(x)\) when we plug \(g(x)\) into \(f(u)\).

Algebraic Manipulation

Algebraic manipulation involves rewriting or simplifying expressions to achieve a particular form. In function composition, such manipulation is often required to split a complex function into a simple composition of two separate functions.
This process entails examining the function’s structure for parts that can be individually analyzed or extracted. In our given function \[ h(x) = (11x^2 + 12x)^2 \], we recognized that by identifying separate roles for the expressions involved, we could distinctively determine:

• The inner function, \( g(x) = 11x^2 + 12x \), as the preliminary transformation.
• The outer function, \( f(u) = u^2 \), designed to perform the final squaring operation.

Through this approach, algebraic manipulation is not only about finding individual component functions but also understanding the operations needed to transform them back into the original function when composed.
Once the functions \(f\) and \(g\) were properly recognized and defined, verifying the composition task became straightforward:Substitute \(g(x)\) into \(f(u)\), rendering \[ f(g(x)) = (11x^2 + 12x)^2\].This confirms the original composition \(h(x)\), showcasing how essential algebraic manipulation is for simplifying and solving function-related problems.