Problem 55 - Intermediate Algebra

Question

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. See Example 5. $$ h(x)=\frac{1}{x-4} $$

Step-by-Step Solution

Answer

Choose \(g(x) = x - 4\) and \(f(x) = \frac{1}{x}\). They satisfy \(h(x) = (f \circ g)(x)\).

Step 1: Understand the Composition

The composition of two functions, denoted as \((f \circ g)(x)\), means that function \(g\) is applied first and then function \(f\) is applied to the result of \(g(x)\). Thus, \((f \circ g)(x) = f(g(x))\). Our task is to break down \(h(x) = \frac{1}{x-4}\) into two such functions.

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

We start by choosing a sensible function for \(g(x)\). To simplify \(h(x)\), we can try \(g(x) = x - 4\). This choice isolates the part \("x - 4"\) from the original function \(h(x)\).

Step 3: Choose a Function for \(f(x)\)

Next, choose \(f(x)\) such that applying \(f\) to \(g(x)\) results in the original function \(h(x)\). Since \(h(x)\) is written as \(\frac{1}{x-4}\), after applying \(g(x)\) we get \(f(g(x)) = \frac{1}{g(x)} = \frac{1}{x-4}\). Hence, we select \(f(x) = \frac{1}{x}\).

Step 4: Verify the Composition

Verify by composing that \((f \circ g)(x) = h(x)\). With \(g(x) = x - 4\) and \(f(x) = \frac{1}{x}\), substituting \(g(x)\) into \(f\) gives: \[f(g(x)) = f(x - 4) = \frac{1}{x-4}\]. This is precisely \(h(x)\), so the decomposition is correct.

Key Concepts

Function Decomposition

Function decomposition is an essential technique in mathematics where a complex function is broken down into simpler parts. In this process, you express a complex function as a composition of two or more simpler functions. For instance, if we have a function \( h(x) \), and we can write it as \( (f \circ g)(x) \), then we are effectively saying \( h(x) = f(g(x)) \).

Breaking down functions into simpler components helps in analyzing and understanding them better. It makes seemingly complicated expressions more manageable and gives insights into their structure or behavior. In this exercise, our task was to decompose \( h(x) = \frac{1}{x-4} \) into two functions, \( f(x) \) and \( g(x) \), where each function embodies a distinct part of the overall behavior of \( h(x) \).

• Start by identifying any transformations in \( h(x) \) that act like a separate function.
• Match these transformations with standard functional forms, like \( g(x) = x - 4 \).
• Next, apply the inverse operation to complete the decomposition, setting \( f(x) = \frac{1}{x} \) after deciding \( g(x) \).

Algebraic Functions

Algebraic functions are functions that are built using simple algebraic operations such as addition, subtraction, multiplication, division, and taking roots. These functions are usually expressed as polynomials, rational functions, and radicals. In our decomposition, the function \( h(x) = \frac{1}{x-4} \) is an example of a rational algebraic function.

Rational functions are formed by dividing two polynomials. The function \( h(x) \) has a numerator of 1 and a denominator \( (x-4) \), showing how it can potentially take on all values except where the denominator is zero.

• The denominator \( x-4 \) indicates a vertical asymptote at \( x = 4 \).
• This asymptote reflects the function's undefined nature at this point, causing a division by zero.

Understanding these foundational traits of algebraic functions supports solving more complex equations and unraveling their intrinsic behavior. It's essential to recognize the form and type of function to apply the right mathematical strategy effectively.

Composition Verification

Composition verification is the step where you ensure that the composed functions reproduce the original function exactly. It's a critical check for accuracy when decomposing functions.

In our example, once we set \( g(x) = x - 4 \) and \( f(x) = \frac{1}{x} \), you need to confirm that the composition \( (f \circ g)(x) \) equals the original function \( h(x) \). This involves substituting the chosen \( g(x) \) into \( f(x) \) to reconstruct \( h(x) \).

• Substitute \( g(x) \) into \( f(x) \), transforming \( f(x) = \frac{1}{x} \) into \( f(g(x)) = \frac{1}{g(x)} = \frac{1}{x-4} \).
• Check if this matches the original function \( h(x) = \frac{1}{x-4} \).

Mismatch at this stage signals errors in identifying \( f(x) \) or \( g(x) \), requiring reevaluation. Correct composition verification confirms the effectiveness of your decomposition, ensuring reliability in problem-solving across various function types.