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Chapter 5: Problem 18

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. \((f-g)(x)\)

Short Answer

Expert verified
(f - g)(x) = x^2 - 2x - 9

Step by step solution

01

Identify the functions

First, identify the given functions: - \(f(x) = x^2 - 9\) - \(g(x) = 2x\) - \(h(x) = x - 3\)
02

Understand the operation

The question asks for \(f - g)(x)\). This means you need to subtract the function \(g(x)\) from the function \(f(x)\).
03

Subtract the functions

To find \(f - g)(x)\), subtract \(g(x)\) from \(f(x)\): \( (f - g)(x) = f(x) - g(x) \).
04

Substitute and simplify

Substitute \(f(x)\) and \(g(x)\) into the equation: \((f - g)(x) = (x^2 - 9) - 2x \).Now, simplify the expression: \( (f - g)(x) = x^2 - 2x - 9 \).

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Key Concepts

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

function subtraction
Subtraction of functions involves subtracting the output of one function from the output of another. To subtract two functions, say \( f(x) \) and \( g(x) \), you perform the operation \( (f - g)(x) = f(x) - g(x) \).
For example, given \( f(x) = x^2 - 9 \) and \( g(x) = 2x \), the subtraction is done as follows:
  • First, identify the individual functions.
  • Next, subtract the expression for \( g(x) \) from \( f(x) \).
  • Ensure you correctly distribute any negatives during subtraction.

    • So, \( (f - g)(x) = f(x) - g(x) = (x^2 - 9) - 2x \).
polynomial functions
Polynomial functions are expressions involving sums of powers of a variable with coefficients. Common examples include linear functions \( ax + b \) and quadratic functions \( ax^2 + bx + c \).
For the given functions:
  • \(f(x) = x^2 - 9\) is a quadratic polynomial because it involves the term \( x^2 \).
  • \(g(x) = 2x\) is a linear polynomial.

When dealing with polynomials:
  • Always line up corresponding powers of \(x\) when adding or subtracting.
  • Simplify by combining like terms.
simplifying expressions
Simplifying algebraic expressions involves combining like terms and performing arithmetic operations to reduce the expression to its simplest form.
For the subtraction \((f - g)(x) = (x^2 - 9) - 2x\), you should:
  • Combine like terms: Notice there are no other \(x^2\) terms or constants to combine.

This results in:
\(x^2 - 2x - 9\).
End result should be neat and in descending order of power terms.
algebraic manipulation
Algebraic manipulation is the process of rearranging and simplifying expressions using various algebraic operations.
Steps to manipulate algebraic expressions:
1. Distribute any multiplication or combine any addition/subtraction.
2. Combine like terms.
3. Organize the expression in descending order of exponent values.

Applied to our example:
Start with:
\((f - g)(x) = x^2 - 9 - 2x\)
Rearrange to get the terms in proper order:
\( x^2 - 2x - 9 \).
You now have a simplified expression from the subtraction of functions!

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