Problem 55 - Elementary and Intermediate Algebra

Question

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. $$ h(x)=\frac{1}{x-4} $$

Step-by-Step Solution

Answer

One possible solution is \(f(x) = \frac{1}{x}\) and \(g(x) = x - 4\).

Step 1: Understand the Composition of Functions

The expression \((f \circ g)(x)\) means that we need to find two functions, \(f(x)\) and \(g(x)\), such that \(f(g(x)) = h(x)\). By identifying \(f\) and \(g\), we will form a composition \(f(g(x))\) that is equivalent to \(h(x) = \frac{1}{x-4}\).

Step 2: Choose a Function for \(g(x)\)

Let's decide to set \(g(x) = x - 4\). This is a common step in function decomposition, where we simplify the expression inside another function. So \(g(x)\) represents the transformation \(x \rightarrow x - 4\).

Step 3: Define the Function \(f(x)\)

Now that we have \(g(x) = x - 4\), we design \(f(x)\) such that \(f(x)\) applied to \(g(x)\) results in \(h(x)\). If \(f(x) = \frac{1}{x}\), then \(f(g(x)) = f(x - 4) = \frac{1}{x - 4}\), which matches \(h(x)\). Thus we define \(f(x) = \frac{1}{x}\).

Step 4: Verify the Composition

To ensure correctness, substitute \(g(x)\) back into \(f(x)\) to check that it equals \(h(x)\): \(f(g(x)) = f(x - 4) = \frac{1}{x - 4}\). This confirms that \(h(x) = \frac{1}{x - 4}\) is formed by the functions \(f(x) = \frac{1}{x}\) and \(g(x) = x - 4\).

Key Concepts

Function Decomposition

Function decomposition is the process of breaking down a complex function into simpler, smaller components. This technique is beneficial when we want to analyze or construct composite functions – those made from combining two or more simpler functions. In our exercise, the function \(h(x) = \frac{1}{x-4}\) needs to be decomposed into two functions: \(f(x)\) and \(g(x)\). The aim is to express \(h(x)\) in terms of these simpler functions, where \(h(x) = (f \circ g)(x)\).

There are some common strategies in function decomposition:

• Identify an inner function \(g(x)\) that represents a simpler transformation or operation.
• Define an outer function \(f(x)\) that directly operates on the output of \(g(x)\).

In this example, \(g(x) = x - 4\) simplifies the expression inside the fraction's denominator, and \(f(x) = \frac{1}{x}\), when applied to \(g(x)\), reconstructs the original function \(h(x)\). This decomposition demonstrates how \(f(g(x))\) results in the complete expression.

Composite Functions

Composite functions are functions composed of two or more functions, structured such that one function's output becomes the subsequent function's input. Written generally as \((f \circ g)(x)\), this means \(f(g(x))\). With function composition, we chain multiple functionalities to create a more complex operation from simpler components.

In our scenario, we aim to express \(h(x) = \frac{1}{x-4}\) as a composite function.

• Set \(g(x) = x - 4\). Here, \(g(x)\) performs a basic transformation.
• Define \(f(x) = \frac{1}{x}\) to evaluate the result of \(g(x)\) – effectively creating the fraction \(\frac{1}{x-4}\).

Using composition, we can simplify complex functions and better understand their components. It also allows for easier manipulations and transformations by focusing on parts of the expression individually.

Function Transformation

Function transformation involves modifying a function's graph or expression's behavior without altering its essential nature. The transformation may involve shifting, stretching, compressing, or reflecting the graph. In our exercise, we explored transformation through function composition.

The function \(g(x) = x - 4\) is a simple transformation; it shifts the graph of \(x\) horizontally by 4 units to the right. This type of transformation is called a translation.

• Translation shifts the graph along the x or y axis. \(g(x)\) demonstrates a horizontal shift to the right.
• \(f(x) = \frac{1}{x}\) reflects the characteristic of a reciprocal transformation, resulting in an inverse-type function.

Combining these concepts, we achieve transformation of \(h(x)\) by composing \(f(x)\) and \(g(x)\), creating a new function that benefits from both shifting and reciprocal qualities. Understanding transformations aids in visualizing not only the algebraic manipulation but also how functions behave graphically with respect to their basic forms.