Problem 19 - 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. $$(g-f)(-2)$$

Step-by-Step Solution

Answer

The result of \\(g-f\\)(-2) is \\(-3\\).

Step 1 - Understand the Operation (g-f)

The expression \(g-f\)(x) represents the function resulting from subtracting \(f(x)\) from \(g(x)\). Formally, this operation is expressed as \(g(x) - f(x)\).

Step 2 - Set Up the Functions

Write down the given functions: \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\).

Step 3 - Subtract the Functions

Perform the subtraction \(g(x) - f(x)\): - Start with \(g(x)\): \(2x - 1\).- Subtract \(f(x)\): \((x^2 + 3x)\).- This results in \(g(x) - f(x) = (2x - 1) - (x^2 + 3x)\).

Step 4 - Simplify the Expression

Simplify \(g(x) - f(x) = 2x - 1 - x^2 - 3x\) by combining like terms:- Combine the \(x\) terms: \(2x - 3x = -x\).- The simplified expression is \(-x^2 - x - 1\).

Step 5 - Compute (g-f)(-2)

Substitute \(-2\) into the simplified function: \(-((-2)^2) - (-2) - 1\).- Calculate \((-2)^2 = 4\).- This gives \(-4 - (-2) - 1\).- Simplify the expression further: \(-4 + 2 - 1 = -3\).

Key Concepts

Function Subtraction

When we talk about function subtraction, we mean taking one function and subtracting its values from another function. This is a basic operation in algebra, similar to subtracting numbers. For instance, if you have two functions \(f(x)\) and \(g(x)\), the subtraction \((g-f)(x)\) is equivalent to \(g(x) - f(x)\). This process allows us to explore the relationships between two different functions and understand how their values compare.
Think of it like this: imagine two journeys taken by two friends; one represented by \(g(x)\) and another by \(f(x)\). To find out how much faster or further one traveled than the other at every point, you would subtract the distances given by \(f(x)\) from those given by \(g(x)\). Subtraction allows us to derive a new function showing this difference over various inputs.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients arranged in terms of powers. A general polynomial function can be expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where the powers are non-negative integers, and the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are real numbers.
Polynomials can take various forms, from simple ones like linear (\(ax + b\)) to more complex ones like quadratics (\(ax^2 + bx + c\)), and beyond. In our exercise, \(f(x) = x^2 + 3x\) is a quadratic polynomial, and \(g(x) = 2x - 1\) is a linear polynomial. These types often appear in real-world problems and are essential in calculus and higher algebra.
Understanding how to manipulate these is crucial, as is the case with subtracting them, which requires careful attention to combining like terms correctly.

Algebraic Expressions

Algebraic expressions form the backbone of algebra. They are made up of numbers, variables, and operations (like addition, multiplication, etc.). Each part of an algebraic expression serves its purpose, whether it's defining the variable's behavior or representing constant values.
When you manipulate algebraic expressions, you often perform operations such as addition, subtraction, and distribution. For instance, in the step-by-step solution, after setting up the functions \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\), we perform subtraction which introduces us to combining like terms.
In subtraction, pay attention to terms with similar variables and corresponding powers because these can be merged. Think of it like gathering apples; group all apples together, and sum them up to simplify your expression.