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Chapter 2: Problem 68

An equation that defines \(y\) as a function of \(x\) is given. (a) Solve for \(y\) in terms of \(x\) and $$\text {replace \(y\) with the function notation } f(x) . \text { (b) Find } f(3)$$. $$-2 x+5 y=9$$

Short Answer

Expert verified
First, isolate y in the equation and then solve for y. Next, replace y with f(x) and find f(3) by plugging 3 in place of x.

Step by step solution

01

Solve for y

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Key Concepts

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

Solving Linear Equations
Linear equations are equations between two variables that produce a straight line graph. To solve a linear equation, we usually aim to isolate the variable of interest on one side of the equation. In this exercise, we start with the given equation:
-2x + 5y = 9.

The goal is to solve for y in terms of x. We start by isolating the term with y on one side:
First, add 2x to both sides of the equation:
  • -2x + 2x + 5y = 9 + 2x
  • 5y = 9 + 2x

Next, we divide each term by 5 to solve for y:
  • y = (9 + 2x) / 5

This expresses y in terms of x. By solving a linear equation in this manner, we can understand the relationship between the two variables and how changing one affects the other.
Function Evaluation
Function evaluation involves finding the value of a function for a specific input value. Given that the relation we found from solving the linear equation is y = (9 + 2x) / 5, we can replace y with the function notation f(x):
  • f(x) = (9 + 2x) / 5

To find f(3), substitute 3 for x in the function:
  • f(3) = (9 + 2(3)) / 5
  • f(3) = (9 + 6) / 5
  • f(3) = 15 / 5
  • f(3) = 3

So, f(3) equals 3. This is the process of function evaluation, which helps us determine the output of a function given a particular input.
Precalculus Methods
Precalculus involves the study of functions and their properties, preparing students for calculus. This exercise incorporates several fundamental precalculus methods:
  • A thorough understanding of linear equations and how to manipulate them.
  • The concept of function notation, where 'f(x)' denotes a function of 'x'.
  • Function evaluation, where specific input values are substituted into the function to find the corresponding output.

By mastering these methods, students build a foundational knowledge that is crucial for more advanced studies in calculus and other areas of mathematics. Precalculus provides the tools and techniques needed to analyze and understand various functions, laying the groundwork for their application in different mathematical and real-world problems.

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Most popular questions from this chapter

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