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Chapter 2: Problem 31

Let \(F(x)=x^{2}-2\) and \(G(x)=5-x .\) Find each of the following. $$(F+F)(1)$$

Short Answer

Expert verified
\( (F + F)(1) = -2 \)

Step by step solution

01

- Understand the function addition

The expression \(F + F\)(x) represents the sum of the function \(F(x)\) with itself. Mathematically, this can be described as \[\(F + F\)(x) = F(x) + F(x).\]
02

- Define the function

Given \[F(x) = x^2 - 2,\] substitute the function expression into the formula from Step 1: \[\(F + F\)(x) = (x^2 - 2) + (x^2 - 2).\]
03

- Simplify the expression

Combine like terms in the expression: \[ (x^2 - 2) + (x^2 - 2) = 2x^2 - 4.\]
04

- Evaluate the function at x = 1

Substitute \( x = 1 \) into the simplified function: \[ \(F + F\)(1) = 2(1)^2 - 4 = 2 - 4 = -2.\]

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Key Concepts

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

Function Operations
When we talk about function operations, we refer to the various ways we can combine or manipulate functions. In this case, we're focusing on addition. Function addition means adding the outputs of two functions. Let's explore the given example where we need to add the function F to itself. We start by writing the expression for function addition, which in our case is:
\( (F + F)(x) = F(x) + F(x) \)
Substituting the value of \( F(x) \), which is \( x^2 - 2 \), we get:
\( (F + F)(x) = (x^2 - 2) + (x^2 - 2) \)
This will lead us to the next core concept: combining like terms.
Combining Like Terms
Combining like terms is an essential algebraic skill. It involves simplifying expressions by merging terms that have the same variable raised to the same power. For the given problem, we have the expression:
\( (x^2 - 2) + (x^2 - 2) \)
We identify similar terms, which in this case are the \( x^2 \) terms and the constant terms. Combining these, we get:
\( x^2 + x^2 - 2 - 2 = 2x^2 - 4 \)
This process helps in simplifying the overall expression, making it easier to work with. Now that we have simplified the function by combining like terms, we can move on to evaluating the function at a specific value of x.
Evaluating Functions
Evaluating functions is the process of finding the output of a function for a specific input. Here, we need to find the value of our simplified function at \( x = 1 \). Our simplified function from the previous steps is:
\( 2x^2 - 4 \)
To evaluate this at \( x = 1 \):
  • Replace every instance of x in the function with 1:
  • \( 2(1)^2 - 4 \)
  • Simplify the expression:
  • \( 2 - 4 = -2 \)
This gives us the final value of \( (F + F)(1) \), which is \( -2 \), effectively concluding the problem.

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Most popular questions from this chapter

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 indicated function values. $$ F(x)=\left\\{\begin{array}{ll}{2 x,} & {\text { if } x<3} \\ {-5 x,} & {\text { if } x \geq 3}\end{array}\right. $$ a) \(F(-1) \quad\) b) \(F(3)\) c) \(F(10)\)

Graph equation after plotting at least 10 points. \(y=\sqrt{x}+1 ;\) use \(x\) -values from 0 to 10

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ 3 x+7=4 $$

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