Why are the concepts of Domain and Range so weird for students to understand?

Students really grapple with the concepts of Domain and Range when dealing with functions. It feels quite disjointed when learning about the properties of certain functions but they are very important concepts that pop up at various points in the A-level curriculum.

Why do they struggle?:

• Poor awareness of definitions

• Poor graph recall

• Inability to link the graph/equation to what domain/range actually is

I have simplified the 2 concepts below:

DOMAIN- What x values can this function accept?

RANGE- What y values does this function output based on the domain?

When you see a question in an A-level paper asking for the domain/range of a function then you need to ask yourself the following: ' what x values can this function accept?' and 'based on what it can accept- what y values does it output?'.

Lets look at some basic examples:

Table of simple examples

Let's start off with the Linear function, here is the graph:

Linear graph example 1

As you can see the graph extends infinitely into the positive x and y axis, there are no apparent restrictions on what x values it can't accept and that means its domain is as follows: All real numbers. It can accept positives/negatives as well as rational/irrational/integers/natural numbers.

In that respect the range is also the same: All real numbers! Based on the domain being all the real numbers that means that the y values will also output positives/negatives as well as rational/irrational/integers/natural numbers.

Let's look at a different graph called the square root:

Square root graph

This particular graph however is not as clear cut. In terms of its domain, lets consider what x values it can accept.

Looking at the graph it seems that you can only substitute values greater than or equal to 0. That makes sense as you cannot square root a negative (unless you do Further Maths but that is a different discussion). The graph seems to increase after 0 and has no other issues in terms of what it can accept. That means the domain is x is greater than 0.

Based on the domain being greater than or equal to 0, our range (y output) looks to be greater than or equal to zero as well. The graph does not show any negative y values in the output which further confirms this. Range: y is greater than or equal to 0.

Final example of this blog concludes with the reciprocal 1/x-2:

Reciprocal graph with asymptote at x=2 (red line added)

This is not clear cut again and invariably students do struggle with reciprocal equations.

Lets start with the domain, its clear to see that the graph is fine with positive and negative values of x but around the 2 on the x-axis we run into an asymptote. This means that the graph is not able to output y values for this x value (think of tan(90) in the same vein).

We can also look at it from the equation itself:

Equation for example 3

If you substituted x=2 into the denominator then you would get 1/0 which is undefined on your calculator. As a result when you get reciprocal questions like this it is worth looking at the denominator first.

Domain: it can accept all real x values but not x=2.

In terms of the Range based on that domain, the graph has another asymptote- the x-axis. If you look at the graph again:

You can see it never touches the x-axis (y=0). Therefore its range is:

Range- All real values of y but not y equal to 0.

TOP TIP: Always write your domain in terms of x and range in terms of y!

In summary, range and domain questions can be tricky but if you work with a clear head and approach them either in a graphical or equation based manner it is entirely possible to get full marks on these questions. It does however depend on you knowing your graphs and also the behaviour of said graphs around various x values.

You also need to look out for the domain given to you in questions, for example:

June 2024 Edexcel Paper 1 - Q8

In this question you can see the domain has been given for both functions f(x) and g(x). It is important you take notice of them as they can impact your range, especially if the domain is limited.

Through constant and purposeful practice you will improve on your ability to determine domain and range. It is an important sub topic as it links to inverse and composite functions as well as parametric equations.