Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(g \circ f)(x)$$

Step-by-Step Solution

Answer

\((g \circ f)(x) = 2x^2 + 6x - 1\)

Step 1: Understand the Composition

When you see \((g \circ f)(x)\), it means you need to evaluate the composition of two functions. In simpler terms, this notation means that you substitute the function f(x) into the function g(x).

Step 2: Substitute f(x) into g(x)

We are given that \(f(x) = x^2 + 3x\). We want to substitute \(f(x)\) into \(g(x) = 2x - 1\)\First, replace every \(x\) in \(g(x)\) with \(f(x)\):g(f(x)) = 2(x^2 + 3x) - 1.

Step 3: Simplify the Expression

Now expand and simplify the expression for \(g(f(x))\):1. Distribute the 2 into both terms of \(x^2 + 3x\):\[2 \times (x^2 + 3x) = 2x^2 + 6x\]2. Subtract 1:\[2x^2 + 6x - 1\]Thus, \(g(f(x)) = 2x^2 + 6x - 1\).

Key Concepts

Substitution in Functions

Function composition is like solving a puzzle where you need to insert one function into another. This process involves something called substitution. When you have two functions, such as \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \), and you're asked to find \((g \circ f)(x)\), this means substituting \( f(x) \) into \( g(x) \).

• Identify the inner function: In this case, that is \( f(x) \). You need to take everything that \( f(x) \) is equal to and plug it into \( g(x) \).
• Replace variables: In \( g(x) = 2x - 1 \), replace every \( x \) with \( x^2 + 3x \), which gives us the new expression \( g(f(x)) = 2(x^2 + 3x) - 1 \).

By understanding substitution, you ensure you're on the right track in combining functions correctly.

Simplifying Expressions

Once you have substituted one function into another, the next step is to simplify the resulting expression. This involves combining like terms and performing any necessary arithmetic operations.Here's how you simplify the expression \( g(f(x)) = 2(x^2 + 3x) - 1 \):

• Distribute constants: Multiply the constant 2 across the terms inside the parentheses: \[ 2(x^2 + 3x) = 2x^2 + 6x \].
• Combine terms: After distribution, simplify further by handling any other operations: \( 2x^2 + 6x - 1 \). This expression is now fully simplified as there are no similar terms to combine, and it reads much cleaner.

Learning how to simplify expressions helps you manage complex equations and makes it easier to handle in further calculations.

Quadratic Functions

Quadratic functions are polynomial functions where the highest degree of the variable is 2. They often appear in the form \( ax^2 + bx + c \). The function resulting from our composition, \( g(f(x)) = 2x^2 + 6x - 1 \), is a quadratic function because the term with the highest power of \( x \) is \( x^2 \).Key characteristics of quadratic functions include:

• Standard form: Quadratics are often expressed in the format \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are coefficients.
• Graphical representation: When graphed, quadratic functions create U-shaped curves known as parabolas. The position and direction of the parabola are determined by the coefficients.
• Applications: Quadratic expressions are frequently used in various real-world scenarios, like projectile motion in physics or calculating areas.

Understanding quadratic functions is crucial for students as they pave the way for deeper mathematical comprehension and real-world problem-solving.