Problem 56 - Elementary and Intermediate Algebra

Question

Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. $$ h(x)=\frac{1}{3 x-16} $$

Step-by-Step Solution

Answer

\(f(x) = \frac{1}{x}\) and \(g(x) = 3x - 16\).

Step 1: Understanding the Composition

The expression \((f \circ g)(x)\) means that the function \(f\) is applied to the result of the function \(g(x)\). Thus, \(h(x) = f(g(x))\). We need to identify possible functions \(f(x)\) and \(g(x)\) that would combine to form the function \(h(x) = \frac{1}{3x - 16}\).

Step 2: Choosing g(x)

To facilitate the expression \(\frac{1}{3x - 16}\), let's assume \(g(x) = 3x - 16\). By making this choice, the composition \(f(g(x))\) becomes \(f(3x - 16)\).

Step 3: Determining f(x)

Since our goal is to have \(f(g(x)) = \frac{1}{3x - 16}\), and knowing \(g(x) = 3x - 16\), we need to find \(f(u)\) where \(u = g(x)\). We set this equal to the form we need: \(f(u) = \frac{1}{u}\). Thus, \(f(x) = \frac{1}{x}\).

Step 4: Verifying the Solution

Substitute \(g(x) = 3x - 16\) back into \(f(x) = \frac{1}{x}\) to ensure that \((f \circ g)(x) = f(g(x)) = f(3x - 16) = \frac{1}{3x - 16}\), which matches \(h(x)\). This confirms that our choice of functions is correct.

Key Concepts

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's like a language where letters and symbols are used to represent numbers and quantities in expressions and equations. When solving algebraic equations, you're often trying to find the value of unknown variables.In the exercise above, you have an algebraic expression for a function given as \( h(x) = \frac{1}{3x - 16} \). To solve this, we need to find two simpler functions, \( f(x) \) and \( g(x) \), that compose to give \( h(x) \). This involves manipulating algebraic expressions and understanding how functions can build upon each other to create more complex functions.

Function Operations

Function operations involve performing basic mathematical operations like addition, subtraction, multiplication, and division on functions. But they also include more complex operations like composition.

• Addition: Given two functions \( f(x) \) and \( g(x) \), their sum is \( (f+g)(x) = f(x) + g(x) \).
• Subtraction: The difference is \( (f-g)(x) = f(x) - g(x) \).
• Multiplication: The product is \( (f \cdot g)(x) = f(x) \cdot g(x) \).
• Division: The quotient is \( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \), provided \( g(x) eq 0 \).

Composition, however, is unique because it involves combining one function with the result of another. In this exercise, we're seeing how the composition \((f \circ g)(x)\) operates by applying \( g(x) \) first and then \( f(x) \). An understanding of function operations helps us manipulate and work with these combinations effectively.

Composite Functions

Composite functions occur when one function is applied to the result of another. In mathematical terms, it's expressed as \((f \circ g)(x) = f(g(x))\). This involves taking an input \(x\), applying the function \(g\), and then applying the function \(f\) to the result of \(g(x)\).In the given problem, we identified \( g(x) = 3x - 16 \) and \( f(x) = \frac{1}{x} \). So, when calculating \( (f \circ g)(x) \), we first use \( g(x) \) to transform the input and then apply \( f(x) \) to get:\[ f(g(x)) = f(3x - 16) = \frac{1}{3x - 16} \]This solution matches our original expression for \( h(x) \), confirming the correctness of our functions. Understanding composite functions is a fundamental skill in algebra, allowing you to break down complex expressions and understand their component parts effectively.