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Chapter 2: Problem 3

Make each of the following statements true by selecting the correct word for each blank. One way to compute \((f-g)(2)\) is to simplify \(f(x)-g(x)\) and then _____ the result for \(x=2\). (evaluate/substitute)

Short Answer

Expert verified
evaluate

Step by step solution

01

Title - Understand the problem

Understand that the task requires selecting the correct word to make the statement true. The mathematical operation involves computing \((f-g)(2)\).
02

Title - Recall the definition of the given functions

Recall that \( (f-g)(x) = f(x) - g(x) \), which means you subtract function \(g(x)\) from function \(f(x)\) for any value of \(x\).
03

Title - Simplify the expression

Simplify the expression \( f(x) - g(x) \) to get a simplified form.
04

Title - Select the correct word

Choose the correct word: after simplifying \( f(x) - g(x) \), to find \( (f-g)(2) \, you must \text{evaluate}\ the result for \ x=2 \). Therefore, the correct word is 'evaluate.'

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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 taking two functions, say \(f(x)\) and \(g(x)\), and creating a new function that represents their difference. The notation for function subtraction is \((f - g)(x)\). This operation is defined mathematically as: \( (f-g)(x) = f(x) - g(x) \).
Here鈥檚 what you need to remember:
  • You're subtracting the entire function \(g(x)\) from \(f(x)\).
  • This subtraction applies to every value of \(x\) for which both functions are defined.

For example, if \( f(x) = 2x + 3 \) and \( g(x) = x^2 \), then:
\( (f-g)(x) = (2x + 3) - (x^2). \)
Sometimes, it helps to think of \(f(x)\) and \(g(x)\) as similar to regular numbers that you subtract from each other, but instead, you're working with entire expressions.
Evaluate Expressions
Once you've simplified the expression from the subtraction of functions, the next step is to evaluate it at a specific value of \(x\). This means substituting the given value of \(x\) into the simplified function and calculating the result.
Here鈥檚 a quick rundown on how to do it:
  • Simplify the expression first.
  • Substitute the given value of \(x\).
  • Calculate the result.

To illustrate, let's continue from our function subtraction example \( (f-g)(x) = 2x + 3 - x^2 \). Suppose we need to find \( (f - g)(2) \).
Step-by-step:
  1. First, simplify the expression: \(2x + 3 - x^2\).
  2. Substitute \( x = 2 \) into the simplified expression:
    \(2(2) + 3 - (2)^2 \).
  3. Calculate the result: \[ 4 + 3 - 4 = 3 \].

Therefore, \( (f - g)(2) = 3 \).
Algebraic Functions
Algebraic functions are mathematical expressions that use a combination of variables (usually \( x \), but any letter can be used) and constants, with operations like addition, subtraction, multiplication, division, and exponentiation.
They form the basis for more complex calculations and transformations in algebra.
Key Points:
  • Functions can be as simple as linear expressions like \( f(x) = 2x + 3 \).
  • They can also be more complex, such as polynomials, quadratic functions, or rational functions.
  • An important part of working with algebraic functions is understanding how to combine and manipulate them, like in function subtraction or addition.

In the previous examples, \( f(x) \) and \( g(x) \) are algebraic functions. Their subtraction \( (f - g)(x) \) and the subsequent evaluation for a specific \(x\)-value illustrate how these operations work in practice.
By mastering these basic operations, you build a strong foundation for tackling more advanced mathematical concepts.

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Most popular questions from this chapter

Combine like terms. [ 1.3] $$a b^{2}+2 a^{2} b-a b^{2}$$

Solve using a graphing calculator. Weekly pay at The Furniture Gallery is \(\$ 450\) plus a \(3.5 \%\) sales commission. If a salesperson's pay was \(\$ 601.03,\) what did that salesperson's sales total?

Combine like terms. [ 1.3] Solve for \(m: x=\frac{m p}{c}\).

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ \frac{1}{2}(x-4)=y $$

For each pair of functions fand g, determine the domain of \(f / g\). $$\begin{aligned} &f(x)=\frac{3}{x-4}\\\ &g(x)=5-x \end{aligned}$$
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