Problem 43 - Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x^{2}-1 .\) Find each of the following. See Example 3 . $$ (g \circ f)(2 x) $$

Step-by-Step Solution

Answer

\( (g \circ f)(2x) = 16x^2 + 8x \).

Step 1: Understand Composite Functions

The notation \(g \circ f\) means the composition of the functions \(g\) and \(f\). It implies that we first use the function \(f(x)\) and then apply the result as an input to the function \(g(x)\). So \( (g \circ f)(x) = g(f(x)) \).

Step 2: Substitute the Expression into the Inner Function

We need to find \(f(2x)\) starting with substituting \(2x\) into \(f(x) = 2x + 1 \). Thus, \(f(2x) = 2(2x) + 1 = 4x + 1\).

Step 3: Use the Result to Substitute into the Outer Function

Now, we substitute \(f(2x) = 4x + 1\) into \(g(x) = x^2 - 1\). Therefore, \(g(f(2x)) = g(4x + 1) = (4x + 1)^2 - 1\).

Step 4: Simplify the Expression

Next, we expand and simplify \( (4x + 1)^2 - 1 \) using algebraic methods: 1. \( (4x + 1)^2 = (4x+1)(4x+1) = 16x^2 + 8x + 1 \).2. Subtract 1: \( 16x^2 + 8x + 1 - 1 = 16x^2 + 8x \).

Step 5: Conclusion - Result of the Composite Function

The final expression of \(g(f(2x))\) is \(16x^2 + 8x\).

Key Concepts

Function Composition

Function composition is all about creating new functions by combining existing ones. When you see notation like \( (g \circ f)(x) \), it signifies that function \( g \) is applied to the result of function \( f \). This means you start with the inside function, \( f \), and feed its output to the outside function, \( g \).

• Start with \( f(x) \). In our case, this is \( f(x) = 2x + 1 \).
• Then substitute the output of \( f(x) \) into \( g(x) \), where \( g(x) = x^2 - 1 \).

This process lets us see how one function outputs new values based on another. It's like following a recipe in steps: first mix the ingredients (\( f(x) \)), then bake (\( g(x) \)).
By understanding this chain of events, function composition becomes a breeze.

Algebraic Simplification

Once you substitute one function into another, as in \( (g \circ f)(x) \), you'll often end up with complex expressions. Simplifying these expressions is essential to get a neat answer. Simplification involves several algebraic steps, such as expanding brackets and combining like terms.
Let's simplify \( (4x+1)^2 - 1 \):

• First, expand \( (4x + 1)^2 \): Multiply \( (4x+1) \) by itself to get \( 16x^2 + 8x + 1 \).
• Next, subtract 1 as per the original expression to simplify it to \( 16x^2 + 8x \).

Expansion and simplification are crucial skills in algebra. They transform messy, complex equations into simple, usable forms. With practice, you'll perform these operations more quickly and accurately, helping you easily solve algebraic problems.

Function Operations

Performing operations involving functions not only encompasses composition but also includes addition, subtraction, multiplication, and division of functions. When dealing with these operations:

• Addition (\( (f + g)(x) = f(x) + g(x) \)) combines the outputs of two functions.
• Subtraction (\( (f - g)(x) = f(x) - g(x) \)) finds the difference.
• Multiplication (\( (f \, \cdot \, g)(x) = f(x) \cdot g(x) \)) produces a product.
• Division (\( \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \)) finds the quotient, provided \( g(x) eq 0 \).

These operations enable us to build new functions from existing ones and analyze functions comprehensively. Being adept at these will enhance your problem-solving skills in algebra by allowing you to work with complex expressions efficiently.