Hello Math Students!
Have you ever asked something like this?
"What is the domain and the range of a function?"
As a math tutor I see students wrestle with this question all the time. Ironically, even many struggling students can answer quickly. They respond:
"The domain is the x and the range is the y."
It's not a bad response. But let's look a little deeper. And let's use an analogy.
In a "mathy" definition, the domain is said to be the set of inputs (usually x-values) that each produce a unique output (y-value) for a function. The range is the resulting set of y-values. From a definition like this, students quickly correlate x-values to the domain and y-values to the range, but their understanding may end at that basic connection, and they may not be able to tackle homework and quiz questions that go any deeper with the concepts.
For an example of a basic domain and range question, consider the doubling function: y = 2x.
WHAT IS THE DOMAIN AND WHAT IS THE RANGE OF THE DOUBLING FUNCTION?
Its domain is the set of all real numbers.
Its range is also the set of all real numbers.
The domain is the set of all real numbers because no matter what number you choose to plug into the doubling function the result is always a single real number. In other words, you can double any real number and get a single real result.
The range is also all real numbers because you can create any real number you want by doubling a number that is half as large.
***Note, zero is in the range too because you can take two times an input (x-value) of zero in the domain to get zero in the range (y-value).
Pretend a function is a math party for numbers!
Now the analogy. Let's pretend a math function is not the usual formula-like expression, but instead let's pretend it's a party where only people (numbers) that are invited can attend. In this comparison the x-values would be like the guest-list for the party.
The doubling function would be a very friendly party because anyone (any number) can come to the party and be "doubled". So the guest list (the domain) is all real numbers.
Now let's look at this more exclusive party. Consider the function y = 2/x
This party does not have a wide open guest list like the doubling function does. Why? Well, the number zero can not come to this party because in math you can't divide by zero and get a real number for a result. In other words, zero would ruin this party. So the domain would be all real numbers except zero!
Try this function out: y = the square root of x
What numbers, if any, would ruin this party?
If you guessed negative numbers, you're right!
In math, you can't take the square root of a negative number and get a real number result.
So the domain (guest list) for this function (party) is all positive numbers and zero, but not any negative numbers.
Expressed with inequalities this same statement would be: x is > or = to 0
If it helps for your future domain and range questions, try thinking of functions as parties for numbers and domains as guest lists for those same parties and see if it helps clarify what is happening.
Stay tuned for more math tips from ATT!