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Chapter 2: Problem 18

Let \(f(x)=-2 x+3\) and \(g(x)=x^{2}-5 .\) Find each of the following. $$(f-g)(x)$$

Short Answer

Expert verified
(f-g)(x) = -x^2 - 2x + 8

Step by step solution

01

- Understand the functions

Given the functions: \(f(x) = -2x + 3\) and \(g(x) = x^2 - 5\).
02

- Define the operation

We need to find \((f - g)(x)\), which means subtracting \(g(x)\) from \(f(x)\).
03

- Substitute the functions

Substitute the given functions into the operation: \((f - g)(x) = f(x) - g(x)\).
04

- Perform the subtraction

Subtract \(g(x)\) from \(f(x)\): \((f - g)(x) = (-2x + 3) - (x^2 - 5)\).
05

- Simplify the expression

Distribute and combine like terms: \((f - g)(x) = -2x + 3 - x^2 + 5\), which simplifies to \((f - g)(x) = -x^2 - 2x + 8\).

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Key Concepts

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

Function Subtraction
Function subtraction involves finding the difference between two functions. It means taking one function and subtracting another from it. In our example, we have two functions: \(f(x) = -2x + 3\) and \(g(x) = x^2 - 5\). To find \((f - g)(x)\), we subtract \(g(x)\) from \(f(x)\). Here鈥檚 how you do it step by step:

  • First, write down the two functions.
  • Next, set up the expression for subtraction: \((f - g)(x) = f(x) - g(x)\).
  • Then, replace \(f(x)\) and \(g(x)\) with their given expressions: \((-2x + 3) - (x^2 - 5)\).
  • Remember that subtracting a function means distributing the negative sign across all terms of the second function (\(g(x)\)).

Now that we鈥檝e set everything up, we can move on to combining like terms.
Combining Like Terms
Combining like terms means gathering terms with the same variables and exponents together. This is an important step in simplifying any expression. Let鈥檚 take our expression from previous step: \((-2x + 3) - (x^2 - 5)\).

  • First, distribute the negative sign through the second function: \(-(x^2 - 5) = -x^2 + 5\).
  • Now, our expression looks like this: \(-2x + 3 - x^2 + 5\).
  • Next, identify the like terms. Here we have: \ \[ -x^2\ \] with no like terms, \ \[ -2x\ \] with no like terms, and constants \ \[3 + 5\ \].

So, we combine the constants together to get 8. The combined expression will now look like: \(-x^2 - 2x + 8\).
Simplifying Expressions
Simplifying expressions means to rewrite the expression in the simplest form possible. It involves combining like terms and making sure the expression is as compact as possible. Let鈥檚 simplify our expression from the previous step: \(-x^2 -2x + 8\).

  • Check each term to ensure it鈥檚 as simple as can be:
  • \

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