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Chapter 5: Problem 79

Given \(f(x)=2 x-3\) and \(g(x)=x 2+3 x-1,\) find the following. $$ (f-g)(x) $$

Short Answer

Expert verified
\((f-g)(x) = -x^2 - x - 2\)

Step by step solution

01

Understand the Problem

We are given two functions, \(f(x) = 2x - 3\) and \(g(x) = x^2 + 3x - 1\), and asked to find \((f-g)(x)\). This means we need to subtract \(g(x)\) from \(f(x)\).
02

Substitute the Functions

Given \(f(x) = 2x - 3\) and \(g(x) = x^2 + 3x - 1\), we need to substitute them into the expression for \((f-g)(x)\), which becomes \((2x - 3) - (x^2 + 3x - 1)\).
03

Distribute the Negative Sign

Subtract \(g(x)\) from \(f(x)\) by distributing the negative sign: \((2x - 3) - x^2 - 3x + 1\).
04

Combine Like Terms

Combine the terms to simplify the expression: \(-x^2 + (2x - 3x) + (-3 + 1)\), which results in \(-x^2 - x - 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Functions
In algebra, a function is like a machine that takes an input, performs specific operations on it, and gives an output. It is expressed as a relation where each input is associated with exactly one output. For example, in the function \( f(x) = 2x - 3 \), the input \( x \) undergoes multiplication and subtraction to yield an output. Similarly, \( g(x) = x^2 + 3x - 1 \) employs squaring, addition, and subtraction. Functions can be represented in multiple ways, including equations, graphs, and tables.
  • The notation \( f(x) \) indicates a function named \( f \) with \( x \) as the variable placeholder.
  • Each function represents a specific rule applied to any value substituted for \( x \).
  • Functions are fundamental concepts in algebra that help model real-world situations.
Subtraction of Functions
Subtraction of functions involves finding the difference between two functions. When given two functions, \( f(x) \) and \( g(x) \), the subtraction \((f-g)(x)\) means subtracting \( g(x) \) from \( f(x) \). To perform this operation:
  • Start by expressing the subtraction as \( (f-g)(x) = f(x) - g(x) \).
  • Substitute the expressions: if \( f(x) = 2x - 3 \) and \( g(x) = x^2 + 3x - 1 \), then \((f-g)(x) = (2x - 3) - (x^2 + 3x - 1)\).
  • Distribute the negative sign across the second function before simplifying, which changes the signs of each term in \( g(x) \).
This process helps to identify the resulting function by showing the mathematical relationship between \( f(x) \) and \( g(x) \) after subtraction.
Combining Like Terms
Combining like terms is a fundamental step in simplifying algebraic expressions. Like terms are terms that share the same variable raised to the same power. For instance, in the subtraction of \( f(x) \) and \( g(x) \) where we arrived at \( -x^2 - x - 2 \), combining like terms involved:
  • Identifying terms with the same variable power, such as \( 2x \) and \( -3x \), which combine to \( -x \) since \( 2x - 3x = -x \).
  • Constant terms like \( -3 \) and \( 1 \) adding up to \( -2 \) following simple arithmetic rules.
Combining like terms allows for a streamlined, simplified expression of the function, making it easier to interpret or use in further calculations. This skill is vital in algebra and can be applied across various types of algebraic problems.

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