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Chapter 10: Problem 7

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=2, \quad P(x)=x-2$$

Short Answer

Expert verified
Yes, \(x=2\) is a zero of the function \(P(x) = x-2\).

Step by step solution

01

Understanding the Problem

The task is to determine if the given value of \(x\) is a zero of the function \(P(x)\). A zero of a function is a value that, when substituted into the function, makes the function equal to zero.
02

Substitute the Given Value of \(x\) into the Function \(P(x)\)

Substitute \(x=2\) into the function \(P(x)=x-2\) to see if the result is zero.
03

Calculate the Function Value

Compute \(P(2)\) by substitution. Replace \(x\) with 2 in the function: \ \(P(2) = 2 - 2\).
04

Simplify the Expression

Simplify the expression: \ \(P(2) = 0\).
05

Determine if \(x=2\) is a Zero

Since \(P(2)=0\), the value \(x=2\) is indeed a zero of the function \(P(x)\).

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Key Concepts

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

Evaluating Functions
In this exercise, we are asked to check if a given value is a zero of a function. Evaluating a function involves plugging a specific value into the function's formula and performing the necessary calculations to determine the function's output. For example, with the function \(P(x)=x-2\), if we want to evaluate \(P(2)\), we substitute 2 into the place of x. Thus, the function becomes \(P(2) = 2 - 2\). This process helps us to understand how functions behave with different inputs and is fundamental in identifying zeros. Remember, the zero of a function is where the function's output equals zero.
Substitution Method
The substitution method is a crucial step in evaluating functions and simplifying expressions. It involves replacing the variable in a function with a given number to see what the result is. For instance, in our exercise, we need to determine if \( x = 2 \) is a zero of the function \( P(x) = x - 2 \). By substituting 2 for x, the function becomes \( P(2) = 2 - 2 \). This substitution process is straightforward but vital to ensuring that the function is evaluated correctly and any zeros are accurately identified. The clear substitution and computation help in verifying whether a particular value satisfies the function.
Simplifying Expressions
Simplifying expressions is the process of reducing complexity in mathematical expressions. After substituting the value into the function, the next step is to simplify the result. For example, in the given function \( P(x) = x - 2 \), after substituting \( x = 2 \), we have \( P(2) = 2 - 2 \). Simplifying this expression gives us \( 0 \). Simplifying is essential because it helps you reach a conclusion about the value you have substituted, determining whether it truly results in zero or not. In this exercise, simplifying correctly confirms that \( x = 2 \) is indeed a zero of the function \( P(x) \). This step ensures clarity and accuracy in mathematical problem solving.

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Most popular questions from this chapter

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{1}{x^{2}}$$

Discuss the possibilities for the roots to each equation. Do not solve the equation. $$x^{4}-x^{3}+x^{2}-x+1=0$$

State the degree of each polynomial equation. Find all of the real and imaginary roots to each equation. State the multiplicity of a root when it is greater than 1. $$x^{4}+2 x^{3}+x^{2}=0$$

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Solve each problem. Average cost of an SUV. Mercedes-Benz spent \(\$ 700\) million to design its new SUV (Motor Trend, www.motortrend.com). If it costs \(\$ 25,000\) to manufacture each SUV, then the average cost per vehicle in dollars when \(x\) vehicles are manufactured is given by the rational function $$A(x)=\frac{25,000 x+700,000,000}{x}$$ a) What is the horizontal asymptote for the graph of this function? b) What is the average cost per vehicle when \(50,000\) vehicles are made? c) For what number of vehicles is the average cost \(\$ 30,000 ?\) d) Graph this function for \(x\) ranging from 0 to \(100,000\).
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