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Chapter 2: Problem 17

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 - 2

Step by step solution

01

- Understand the problem

We are given two functions, \(f(x) = -2x + 3\) and \(g(x) = x^2 - 5\). We need to find \((f + g)(x)\), which represents the sum of the two functions.
02

- Write the formula for \((f+g)(x)\)

The sum of two functions \(f\) and \(g\) is given by \((f+g)(x) = f(x) + g(x)\). We need to add \(f(x)\) and \(g(x)\) together.
03

- Substitute the given functions

Substitute the given expressions for \(f(x)\) and \(g(x)\):\[ (f+g)(x) = (-2x + 3) + (x^2 - 5) \]
04

- Combine like terms

To combine the terms, add the linear terms together and add the constant terms together: \[ (f+g)(x) = x^2 - 2x + 3 - 5 \]. Simplify further to get \[ (f+g)(x) = x^2 - 2x - 2 \]

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Key Concepts

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

function operations
Function operations involve performing arithmetic operations like addition, subtraction, multiplication, and division on functions. In this exercise, we focus on addition.
When adding two functions, you combine them by adding their respective outputs for the same input, which means we sum the expressions of the functions after substitution.

For example, given two functions:
  • For f(x) = -2x + 3
  • For g(x) = x^2 - 5
The sum of these functions is given by the expression (f+g)(x) = f(x) + g(x).
To find this, you substitute the given functions and add their outputs for any value of x.
polynomial functions
Polynomial functions are a combination of variables and coefficients involving only addition, subtraction, and multiplication, and non-negative integer exponents of variables. Each term in a polynomial is called a monomial.
For example, consider our given functions:
  • f(x) = -2x + 3 is a linear polynomial function because the highest power of x is 1.
  • g(x) = x^2 - 5 is a quadratic polynomial function because the highest power of x is 2.
When we add these two functions, we are essentially combining two polynomial expressions. The result will be a new polynomial function with the highest degree of the involved terms, which in this case is degree 2.
combining like terms
Combining like terms is a process used to simplify polynomial expressions by adding or subtracting terms that have the same variables raised to the same power. This process makes the expression simpler and easier to work with.
In our exercise, to combine like terms, we look at the polynomial expression:
  • (f+g)(x) = (-2x + 3) + (x^2 - 5)
First, distribute the terms:
  • x^2 + (-2x) + 3 + (-5)
Next, combine like terms (terms with the same variable and exponent):
  • x^2 (no like terms to combine it with)
  • -2x (no like terms to combine it with)
  • 3 and -5 (both constant terms)
So, (f+g)(x) combines to give: x^2 - 2x - 2.

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Most popular questions from this chapter

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graphing calculator to check each prediction.

Explain why the domain of the function given by \(f(x)=\frac{x+3}{2}\) is \(\mathbb{R},\) but the domain of the function given by \(g(x)=\) \(\frac{2}{x+3}\) is not \(\mathbb{R}\)

Find the intercepts. Then graph by using the intercepts, if possible, and a third point as a check. $$ 0.2 y-1.1 x=6.6 $$

Graph equation after plotting at least 10 points. \(y=x^{3}-5 ;\) use \(x\) -values from \(-2\) to 2

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge. Cal's Parking charges \(\$ 5.00\) to park plus \(50 \mathrm{c}\) for each \(15-\mathrm{min}\) unit of time. Estimate how long someone can park for \(\$ 9.50 .\)
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