Month: July 2023


Jul
2023

Teaching Inverse Functions in Algebra

categories: Teaching Secondary Math
Teaching Inverse Functions

In elementary school, students learn how some mathematical operations are opposites of other operations. Addition is the opposite of subtraction. Multiplication is the opposite of division. As students get older, we introduce the word inverse. Inverse operations undo what has been done and are essential for solving equations. In algebra we expand the idea of inverses to functions. In geometry, the inverse of a conditional statement is when the hypothesis and conclusion are negated. In calculus students learn that integration is the inverse of taking the derivative.

Today we are going to delve into inverses with respect to the algebraic lens of functions, both analytically and with the geometric connection graphically.

Graphical Inverse

A function and its inverses are connected graphically by being the reflection over the line y=x.

When I start teaching inverses of functions, I often will start with points on the coordinate plane and have students switch the x and y coordinates to find the inverse. If you do this with several points and then ask students to graph the line y=x, they often will notice the connection of the reflection over the line y=x on their own. Here is a sample prompt to give your students.

We would discuss the domain and range of the function and its inverse as well as deciding if the inverse is still a function or just a relation.

Next, we look at a linear function, because this is the function that my students are most comfortable with and you do not need to restrict the domain for the inverse to be a function.
I may use a function similar to the graph shown in the first image above, f(x)=2x+4. Students will table the function and then switch the x and y coordinates to graph the inverse. They then would graph the line y=x and really see the connection that the function and its inverse are reflected over the line y=x. It would be good to note whether the inverse of the function is also a function at this point, as it will set up a future discussion when graphing parabolas.

After students start to get a feel for what an inverse looks like graphically, it is time to introduce the algorithm for finding the inverse.

Steps to Find the Inverse of a Function Algebraically

We will go over the steps and an example followed by some practice problems.

Restricted Domains

Finally, we will look at functions that need to have a restricted domain to make the function’s inverse also a function. It should be noted that if the inverse is not a function, then we would classify it simply as a relation.

The parent function for quadratics is a great function to start with when looking at restricted domains.

How do you teach inverse functions? Leave a comment below! If you are interested in my lesson or more teaching resources for inverse functions, visit my TPT store below!


Jul
2023

Teaching Interval Notation

categories: Key Features of Functions

Interval notation is a way to describe a set of numbers. In elementary school, students are introduced to number lines and inequalities. Number lines are a wonderful visual tool for students to make sense of numbers and to process whether a value is greater than or less than another value.  Eventually, these ideas morph into a solution set when they reach pre-algebra. For example, x is greater than 3 has infinitely many solutions. Its solution set could be displayed on a number line with an open ended ray.

Students first learn to write this analytically as an inequality, x > 3. We can write this same interval of values with interval notation such as (3,∞). In algebra, we start to study intervals of numbers such as the domain and range. Instead of using an inequality to represent an interval of real numbers, interval notation is often used instead.  When a set of numbers does not include the endpoint, as shown above, a parenthesis is used to indicate that the interval approaches that number, but does not include it in the interval.

Suppose we would like to include the endpoint of an interval. Let’s look at the interval when x is less than or equal to 2.

As an inequality we would write this as x ≤ 2.  However as an interval of values, we would use a square bracket to show that the endpoint 2 is included in the interval such as (-∞, 2]. Note that one can never actually reach infinity, so infinity will always have a parenthesis and not a square bracket.

Sometimes an interval of numbers has a starting and ending point on the number line. This is sometimes referred to as an “and” compound inequality.

As an inequality we would write -3<x<1. And then for interval notation it would look like (-3,1).

We can make a union of intervals when the rays go in opposite directions. This would be an “or” compound  inequality.

The set of all real numbers can also be written with interval notation.

A common mistake I students make when using interval notation is to write the larger value first and then the smaller value. So, make sure that when you introduce interval notation to your students that you remind them it looks like (lower bound, upper bound).

You can grab my free interval notation graphic organizer here. I also have a full version Interval Notation Lesson available to purchase in my TPT store that is part of my Key Features Unit.