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Chapter 2: Problem 24

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

Short Answer

Expert verified
The value of \( (F-G)(2) \) is -1.

Step by step solution

01

Understand the Problem

The problem requires the evaluation of \(F-G\) at \(x = 2\), where \(F-G\) represents the difference between the functions \(F(x)\) and \(G(x)\).
02

Write the Expressions for F(x) and G(x)

Given \(F(x) = x^{2} - 2\) and \(G(x) = 5 - x\).
03

Define (F-G)(x)

\((F-G)(x) = F(x) - G(x) \). This translates to \((F - G)(x) = (x^{2} - 2) - (5 - x) \).
04

Simplify the Expression

Simplify \((x^{2} - 2) - (5 - x)\) to \(x^{2} - 2 - 5 + x = x^{2} + x - 7\). Therefore, \((F - G)(x) = x^{2} + x - 7\).
05

Evaluate (F-G)(2)

Substitute \x = 2\ in the simplified expression: \(2^{2} + 2 - 7\).
06

Simplify the Result

Calculate \(4 + 2 - 7 = -1\).

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Key Concepts

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

Polynomial Functions
Polynomial functions are a type of algebraic expression that includes terms made up of a variable raised to a non-negative integer exponent. They look like this:
  • Linear polynomials: e.g.,\(2x + 3\)
  • Quadratic polynomials: e.g.,\(x^2 - 4x + 4\)
  • Cubic polynomials: e.g.,\(x^3 + 2x^2 - x + 1\)
Understanding polynomial functions helps in solving algebra problems like the one you're working on. Each polynomial term can contribute differently to the overall shape and behavior of the function graph.
Function Evaluation
Function evaluation is the process of finding the value of a function at a specific input. For example, to find \(F(2)\) for the function \(F(x) = x^2 - 2\), you substitute \(x = 2\) into the function.

Here's how we do it step-by-step:
  • Identify the function, e.g., \(F(x) = x^2 - 2\)
  • Substitute the given value of \(x\), which is 2 in this case, into the function: \(F(2) = 2^2 - 2\)
  • Calculate the result: \(F(2) = 4 - 2 = 2\)
Function evaluation is crucial because it helps us understand the behavior of functions at specific points.
Simplifying Expressions
Simplifying expressions involves combining like terms and performing arithmetic operations to make an expression as compact and understandable as possible. Let's break down the process using your problem as an example.
  • Given functions: \(F(x) = x^2 - 2\) and \(G(x) = 5 - x\)
  • To find \((F-G)(x)\), follow the equation: \((F-G)(x) = F(x) - G(x)\)
  • Substitute \(F(x)\) and \(G(x)\) into the equation: \((F - G)(x) = (x^2 - 2) - (5 - x)\)
  • Remove the parentheses carefully: \(x^2 - 2 - 5 + x\)
  • Combine like terms: \(x^2 + x - 7\)
This simplified form makes further calculations, such as finding specific function values, much easier.

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Most popular questions from this chapter

Find the intercepts. Then graph by using the intercepts, if possible, and a third point as a check. $$ 5 y-x=5 $$

Find the indicated function values. $$ f(x)=\left\\{\begin{array}{ll}{x,} & {\text { if } x<0} \\ {2 x+1,} & {\text { if } x \geq 0}\end{array}\right. $$ a) \(f(-5) \quad\) b) \(f(0)\) c) \(f(10)\)

Find the domain of the function given by each equation. $$ g(x)=\frac{1}{2} x $$

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

Find the slope of the line that contains the given pair of points. a) \((5 b,-6 c),(b,-c)\) b) \((b, d),(b, d+e)\) c) \((c+f, a+d),(c-f,-a-d)\)
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