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Chapter 9: Problem 42

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

Short Answer

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Step by step solution

01

Understand the Problem

We need to find \(f-g\) evaluated at \(-3\). This is calculated by \(f(-3) - g(-3)\).
02

Evaluate f(-3)

Given \(f(x)=x^{2}-9\), substitute \x = -3\ into the function: \[ f(-3) = (-3)^{2} - 9 = 9 - 9 = 0 \]
03

Evaluate g(-3)

Given \(g(x) = 2x\), substitute \x = -3\ into the function: \[ g(-3) = 2(-3) = -6 \]
04

Calculate the Difference

Using the obtained values, compute \(f(-3) - g(-3)\): \[ (f-g)(-3) = f(-3) - g(-3) = 0 - (-6) = 0 + 6 = 6 \]

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Key Concepts

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

Function Evaluation
Function evaluation is the process of finding the output of a function for a specific input value. In this exercise, we evaluated the functions f(x) and g(x) at x = -3. Evaluating a function often involves simple substitution and arithmetic. It鈥檚 important to carefully follow each step to avoid errors.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations. They can be used to represent functions. For instance, the functions given in the exercise are represented as algebraic expressions:
  • For f(x)=x^{2}-9, the square of x is subtracted by 9.
  • For g(x)=2x, x is multiplied by 2.
Understanding how to manipulate these expressions is essential in solving function operation problems.
Substitution Method
The substitution method involves replacing a variable with a given number to find the value of an expression or function. In the exercise, we substituted
  • -3 for x in f(x) to get f(-3)
  • -3 for x in g(x) to get g(-3).
By substituting the values correctly and simplifying the expression, we found that (f-g)(-3) = 6. This method is helpful in various algebra problems, allowing us to find specific outputs based on given inputs.

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Most popular questions from this chapter

Solve each problem. The weight of a bass varies jointly as its girth and the square of its length. (Girth is the distance around the body of the fish.) A prize-winning bass weighed in at \(22.7 \mathrm{lb}\) and measured 36 in. long with a 21 -in. girth. How much (to the nearest tenth of a pound) would a bass 28 in. long with an 18 -in. girth weigh?

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ h)(-2) $$

Let \(f(x)=x^{2}\) and \(g(x)=2 x-1 .\) Match each expression in Column \(I\) with the description of how to evaluate it in Column II. II A. Square \(5 .\) Take the result and square it. B. Double 5 and subtract \(1 .\) Take the result and square it. C. Double 5 and subtract 1 . Take the result, double it, and subtract 1 . D. Square \(5 .\) Take the result, double it, and subtract 1 . I $$ (f \circ g)(5) $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=-x $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{g}{h}\right)(-1) $$
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