Chapter Three - Functions vs. Equations - Using the Toolbox - Learn to solve for x

Oct 16, 2026

Chapter Three explains the key difference between functions and equations by contrasting how a function assigns outputs to chosen input values with how an equation asks which value of x satisfies a relationship. It walks through reasoning steps for solving equations, emphasizing going backwards and applying inverse operations. The chapter introduces a “toolbox” of equation rules and demonstrates their systematic use to isolate x.

LanguageEnglish

CategoryExam

FormatPDF

Pages28

Reading Time1 h 11 min

AccessFree to download

Chapter Three: Functions vs. Equations

Functions vs. Equations

Using the Toolbox

Chapter Three Functions vs . Equations Functions vs. Equations At first look, it may seem that functions and equations are the same thing. They are both statements that have an equal sign. Let’s look closely. Here is a function: f (x) = x 4 − 5x3 + x , and here is an equation: 52 = x 2 + 3 x . The first example is a function. f is telling the x that is in the slot what to do, in this case, multiply by itself four times, subtract five times that x-value that is multiplied by itself three times, and finally adding the square root of itself. Specifically, if x = 2 , then f (2) = (2)4 − 5(2)3 + 2 which is approximately 16 − 5(8) + 1.414, or 24.414. We started with x = 2, and got the result 24.414. We can take other values for x, substitute them into the slot, and get other numbers, and there is an infinite number of values to choose. An equation, however, does not have a slot. There is nothing telling x what todo. Instead, we are faced with the question “what is x?” It is the obvious question with our above-given question,“what x-value works, in that when I take x, square it, add three times the square root of x, that I get 52?” It turns out that x is approximately equal to 6.65. There is only one x-value that works. In getting 6.65, we have solved for x. This is the difference between functions and equations: with functions we have aslot, we have many options for what x can be, we can graph the relation between what x is and what the result for each x is, there is a height associated with each xvalue, and we could go on about domain and range and all the things we talked about in chapter one. But when we are faced with an equation, we cannot just plugin any x; we must go backwards and try to determine a unique and, in most cases singular, answer. There is no “plugging in” with an equation. Instead, we solve for x. Here is a picture of what I’m talking about. Let f (x) = 2x − 4 , and we can plot a graph: However, if I simply write the equation that works, and that is x = 6 . 8 = 2x − 4 , there is only one x-value The rest of this chapter will be devoted to learning ways to solve for x. We will be working with equations, and not functions. Using the Toolbox In the beginning of the book, I mentioned that there are two kinds of ways todo mathematics. One involves the “tattered cookbook” approach, in which one has hundreds of recipes to pick from to solve the problem. Each is laid out with the proper materials (ingredients) and then directions. Follow the directions and you successfully prepare the meal item. People who are successful with doing mathematics this way are successful only because they have memorized the various many “recipes” through great practice, they have memorized the tricks, and the worst part of all of that is that sometimes they may not know why they do what they do. A plumber, however, has been taught through great practice to look for the symptoms of the problem and to solve the problem with the few tools that he brings with him. He understands the potential weakness within the system, understands what causes the leak, and knows what tools to use and how to apply them. He doesn’t bring a manual with him to refer to as he assesses the problem; he sees it, knows what to do from great practice, and uses the tools that he has at hand to solve the problem. He knows how to use the tools from great practice, reaches for that one tool each time that he needs, and always brings these tools when approaching the problem. In the equation 4x + 3 = 7 , the general technique to solve is to first subtract 3 from both sides. These words may well have entered into your mind as you looked at it, and you jump right in and do it. Why?“Well, that’s what I was taught to do first!” Good, but eventually this kind of reasoning will lead to trouble, because as you advance there may be steps or tools that are not in your tattered cookbook. Try thinking this way: If 4x + 3 = 7 , then 4x + 3 − 3 =

FAQs

What is the main difference between a function and an equation?

A function has an input-output slot, so you choose x and compute the result. An equation has no slot; instead, you solve for the x-value that makes the statement true.

Why do you “go backwards” when solving an equation?

Because solving for x requires undoing operations applied to x. Each step applies an inverse move (e.g., add to undo subtract, divide to undo multiply) to isolate x.

What is the purpose of the “toolbox” in solving equations?

The toolbox lists logical if-then rules that justify each transformation on both sides of an equation. Using them step-by-step helps you simplify the equation while keeping it equivalent.

Oct 16, 2026

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