Problem 7 - 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. $$(f \circ g)(3)$$

Step-by-Step Solution

Answer

\((f \circ g)(3) = 40\).

Step 1: Understand the Composition

The notation \((f \circ g)(x)\) means that we first apply the function \(g(x)\) and then apply the function \(f(x)\) to the result.

Step 2: Apply Function g

First, substitute \(x = 3\) into \(g(x)\). Therefore, \(g(3) = 2(3) - 1 = 6 - 1 = 5\).

Step 3: Apply Function f

Now use the result from Step 2 and substitute \(g(3) = 5\) into \(f(x)\). Thus, \(f(g(3)) = f(5)\).

Step 4: Calculate f(5)

Substitute \(x = 5\) into \(f(x) = x^2 + 3x\). Thus, calculate \(f(5) = (5)^2 + 3(5) = 25 + 15 = 40\).

Step 5: Conclude the Composition

Therefore, \((f \circ g)(3) = f(g(3)) = f(5) = 40\).

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole number powers and have constant coefficients. These functions are fundamental in algebra and show up frequently across math subjects. Let's take a look at some important characteristics of polynomial functions:

• Terms: A polynomial is made up of terms, each being a product of a constant and a variable raised to a power. For example, in the polynomial function \( f(x) = x^2 + 3x \), there are two terms: \( x^2 \) and \( 3x \).
• Degree: The degree of a polynomial function is the highest power of the variable in the expression. In \( f(x) = x^2 + 3x \), the degree is 2, because the highest power of \( x \) is 2.
• Operations: Polynomials can be added, subtracted, and multiplied. In our scenario, we are using the composition of functions to evaluate a polynomial function at a certain point.

Understanding polynomial functions is important because they serve as the building blocks for more complex expressions. Knowing how to manipulate them can simplify many mathematical tasks.

Function Evaluation

Function evaluation involves finding the value of a function given a specific input. It's like asking, "What number comes out when I put this number in?" Let's break down how function evaluation works:

• Substitution: To evaluate a function, replace the variable with the number you are given. For instance, if you need to find \( f(3) \) for \( f(x) = x^2 + 3x \), substitute \( 3 \) in place of \( x \).
• Calculating Values: After substitution, compute the expression. Using the example above, \( f(3) = 3^2 + 3(3) = 9 + 9 = 18 \).
• Composition: Sometimes you evaluate one function and use its result in another function, like in this exercise. Here, we first evaluate \( g(3) \) to obtain the result needed for \( f(g(3)) \).

The key is to follow the operations in the correct order, especially in compositions, ensuring one function's output becomes another function's input seamlessly.

Mathematical Operations

Mathematical operations are the actions or processes that change or combine numbers and variables. They form the core of algebraic problem-solving and are essential for function composition. Here's a bit more on how they work:

• Basic Operations: Include addition, subtraction, multiplication, and division. In the function \( g(x) = 2x - 1 \), multiplication is used to compute \( 2x \), and subtraction is used to find \( 2x - 1 \).
• Order of Operations: Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures that calculations are performed correctly. In \( f(x) = x^2 + 3x \), exponents are calculated before addition.
• Composition of Functions: Here, the output of one function becomes the input of another. This requires careful application of operations to maintain correct sequence. We first calculate \( g(3) \), before using that output in \( f(x) \).

Understanding these operations and applying them correctly is vital to mastering algebra and functions. They allow you to manipulate equations skillfully, leading to accurate solutions.