We explore the similarities and differences between expressions, equations, and functions. We also look at when it is and is not appropriate to use the equals sign.

Introduction

Now that we are exploring the zeros of functions, one issue that often comes up for students (and for teachers reading students’ work!) is when you should and should not use an equals sign. We are going to review when it is and is not ok to use an equals sign.

First, it is helpful to review a few important terms. In mathematics, it is important to use these and other terms precisely so that you are communicating clearly and saying what you intend to say. Speaking precisely using mathematical terms can be difficult to learn and takes some practice!

Expressions

Here are some examples of algebraic expressions:

The important thing to notice is that there are no equals signs in an expression. There are also no inequality signs.

Here are some examples of mathematical expressions:

Every algebraic expression is also a mathematical expression.

Here is an example of evaluating an expression. Consider the expression . Let’s evaluate that expression at .

Notice that when evaluating this expression at a particular point, we can use an equals sign. This is a good use of the equals sign and shows us simplifying. But, we should not put an equals sign between and as these two expressions are only equal when .

Equations

When we use an equals sign to say that two different mathematical expressions give the same value, we are creating an equation.

Here are some examples of equations:

When we are given an equation in a problem, we often want to know what value of the variable will make the equation true. That is, what value of the variable will make both sides give the same value.

Here is an example of solving an equation. Let’s solve .

Notice that this is the reverse process of evaluating the expression . When evaluating the expression, we knew the -value and substituted it in. When solving the equation, we knew what the output should be and had to find the -value that would produce that output. In fact, we found two such values!

This is a key observation. Notice that if we wrote we would be saying something not true. In particular, we would be claiming that !

The best thing is to do when solving an equation is to make a new line for each step, but if you need to write your steps on a single line, you can use an arrow to show the next step. For example, we could write We could also use connecting words between equations. For example, we could write:

Zeros of Functions Revisited

Notice that when we are working with functions, we are also working with equations and expressions.

  • When we write , we are referencing the function by name.
  • When we write , this is an expression for the output of the function at .
  • When we write , we are defining the way the function produces outputs.
  • When we want to find the zeros of this function, we set up the equation . In our case, this would mean solving the equation . Notice that this is solving an equation so we do not write equals signs between the steps.

Another important connection between finding zeros of functions and solving equations is that every equation can be thought of as the zero of a function. Consider the following example.