Problem 92 - A Graphical Approach to College Algebra

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 \(f(x) = x^2\) and \(g(x) = 11x^2 + 12x\).

Step 1: Identify the Target Function

We start with the given function \(h(x) = (11x^2 + 12x)^2\). Our task is to express this function as a composition of two functions \(f\) and \(g\) such that \((f \circ g)(x) = h(x)\). This means \(h(x) = f(g(x))\).

Step 2: Choose the Inner Function

To simplify the problem, choose \(g(x) = 11x^2 + 12x\) since it matches the expression inside the square in \(h(x)\). This means \(g(x)\) should be the expression inside the parenthesis.

Step 3: Define the Outer Function

Since choosing \(g(x) = 11x^2 + 12x\) results in \(h(x) = (g(x))^2\), we define \(f(x) = x^2\). This function takes the result of \(g(x)\) and squares it, replicating the original \(h(x)\).

Step 4: Verify the Function Composition

Check that \((f \circ g)(x) = f(g(x)) = f(11x^2 + 12x) = (11x^2 + 12x)^2 = h(x)\). This confirms that our chosen functions \(f\) and \(g\) work correctly to represent \(h(x)\) as \(f(g(x))\).

Key Concepts

Inner Function

In the exercise, we need to break down the function \(h(x) = (11x^2 + 12x)^2\) into a composition of two functions. This involves identifying an inner function, denoted as \(g(x)\), through which the rest of the function will be processed.

The key to choosing the inner function is to look inside the outer structure of \(h(x)\). By identifying \(g(x) = 11x^2 + 12x\), we isolate the expression inside the parentheses. This inner function is a crucial step for simplifying and managing complex equations.

• Identifying an inner function simplifies the main problem.
• Think of it as the core component that will be manipulated externally by another function.

Remember, the purposeful selection of \(g(x)\) paves the way for a successful composition with the outer function.

Outer Function

Once the inner function \(g(x) = 11x^2 + 12x\) is determined, the next step is to define the outer function \(f(x)\). The role of the outer function is to transform the outcome of the inner function to match the original function \(h(x)\).

In this scenario, we choose \(f(x) = x^2\) because it nicely complements \(g(x)\). The outer function takes the result from \(g(x)\) and applies the operation needed to retrieve \(h(x)\), which in this case is squaring the output of \(g(x)\).

• An outer function defines how the inner function is processed further.
• It often involves operations like exponentiation, as we see here with squaring.

By applying \(f(x)\) to \(g(x)\), we can systematically rebuild the original function, ensuring that \((f \circ g)(x) = h(x)\).

Algebraic Functions

Algebraic functions include operations such as addition, subtraction, multiplication, division, and exponentiation involving variables and constants. The functions \(h(x) = (11x^2 + 12x)^2\), \(f(x) = x^2\), and \(g(x) = 11x^2 + 12x\) are prime examples of such functions.

These algebraic components often need to be combined, as seen in composition, where one function is applied to the output of another. Understanding the individual algebraic components helps simplify complex problems into manageable steps.

• Algebraic functions map inputs directly to outputs using algebraic expressions.
• Function composition uses these inputs and outputs creatively to form new functions.

By analyzing each algebraic component, the overall behavior of the composite function can be easily understood and applied, as demonstrated in this exercise.