Problem 18 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

(f-g)(-2) = 3

Step 1: Understand the Expression

We are given the task to find \((f-g)(-2)\). This means we have to first calculate \((f-g)(x)\) and then evaluate it at \(x = -2\).

Step 2: Define the Function Difference

The expression \((f-g)(x)\) represents the difference between the functions \(f(x)\) and \(g(x)\). Let's compute it: \((f-g)(x) = f(x) - g(x)\).

Step 3: Substitute the Function Definitions

Substitute the given functions into the difference: \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\). So, \((f-g)(x) = (x^2 + 3x) - (2x - 1)\).

Step 4: Simplify the Expression

Simplify the expression by distributing and combining like terms: \((f-g)(x) = x^2 + 3x - 2x + 1 = x^2 + x + 1\).

Step 5: Evaluate the Function at x = -2

Now, substitute \(x = -2\) into the simplified expression: \((f-g)(-2) = (-2)^2 + (-2) + 1\).

Step 6: Simplify the Evaluation

Calculate the expression: \((-2)^2 = 4\), \(-2 + 1 = -1\). So, \(4 - 1 = 3\).

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. In essence, you insert the output of one function into the input of another. For example, if you have functions \( f(x) \) and \( g(x) \), the composition \( (f \circ g)(x) \) means you first put \( x \) into \( g \), and then take the result to plug into \( f \). This results in a completely new function that captures a series of steps.

Here's a simple breakdown of function composition process:

• Evaluate the inner function \( g(x) \) first. This gives you a new expression or value.
• Use the result from \( g(x) \) as the input for \( f(x) \), creating the composite function \( f(g(x)) \).

Function composition is quite different from just evaluating a single function, as it combines two operations in sequence.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

Expressions can be simple, such as \( 3x \), or more complicated, like \( x^2 + 3x - 2x + 1 \). In algebra, we often simplify expressions. This involves combining like terms and performing basic arithmetic to make the expression more concise.

For example, when simplifying \((f-g)(x) = (x^2 + 3x) - (2x - 1)\):

• Distribute any negative signs through the expression.
• Combine like terms (those that have the same variable and exponent).
• This results in a simplified version of the expression.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input value. Essentially, we're seeking what the function equals when it operates on a certain value of \( x \).

To evaluate a function, follow these steps:

• Take the simplified function expression, such as \( f(x) = x^2 + x + 1 \).
• Substitute the specific value of \( x \) (like \(-2\) in our exercise).
• Perform the arithmetic operations to find the result.

So, the importance of function evaluation hinges on accurately calculating the result for the given input, ensuring you follow all arithmetic rules correctly.

Polynomial Functions

Polynomial functions are algebraic expressions consisting of terms of variables raised to whole number exponents, where the highest exponent dictates the degree of the polynomial.

For example, with \( f(x) = x^2 + x + 1 \):

• The term \( x^2 \) signifies we have a quadratic polynomial, as the highest degree is 2.
• Lower degree terms in the polynomial, like \( x^1 \) or constant terms, also make up the polynomial.
• Polynomials can be categorized not just by degree, but also by their number of terms (e.g., binomial, trinomial).

Understanding polynomials is vital because they form the foundation of many algebraic concepts and operations such as factoring, finding roots, and graphing functions.