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**Common Algebra Question...What's the domain and the range? ANSWER: Pretend a function is a math party!**

August 19, 2019

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Hello Math Students!

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Have you ever asked something like this?

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"What is the domain and the range of a function?"Â

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As a math tutor I see students wrestleÂ with this question all the time. Ironically, even manyÂ strugglingÂ students can answer quickly. TheyÂ respond:

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"The domain is the x and the range is the y."

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It's not a bad response. But let's look a little deeper. And let's use an analogy.

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In a "mathy" definition, the domain is said to be the set ofÂ inputs (usually x-values) that each produce a unique output (y-value)Â for a function. The range isÂ the resulting set of y-values. From a definitionÂ like this, students quickly correlate x-values to theÂ 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.

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For an example of a basic domain and range question, considerÂ the doubling function: y = 2x.

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QUESTION:

WHAT IS THE DOMAIN AND WHAT IS THE RANGE OF THE DOUBLING FUNCTION?

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ANSWER:

ItsÂ domain is the set of allÂ real numbers.

Its range is also the set of all real numbers.

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EXPLANATION:Â

TheÂ domain is the set of allÂ real numbersÂ because no matter what number you choose to plug into the doublingÂ function the result is always a single real number. InÂ other words, you canÂ double any real number and get a singleÂ real result.

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The range is also all real numbersÂ because you can create any real number you want by doubling a number that is half as large.

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***Note, zero is in the range too because you can take two times an inputÂ (x-value) ofÂ zero in the domain to get zero in the range (y-value).

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Pretend a function is a math party for numbers!

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Now the analogy.Â Let's pretend a math function is not the usual formula-like expression, but instead let's pretend it's aÂ party where only people (numbers) that are invited can attend.Â In this comparisonÂ the x-values would be like the guest-list for the party.

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The doubling function would be a very friendly party because anyone (any number) can come to the party and be "doubled". SoÂ the guest list (the domain) is all real numbers.

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Simple, right?

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Now let's look at this more exclusive party. Consider the function y = 2/x

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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 mathÂ you can't divide by zero and get a real number for a result. In other words, zero would ruin this party. SoÂ the domain would be all real numbers except zero!

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Try this function out: yÂ = the square root of x

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What numbers, if any, would ruin this party?

If you guessed negative numbers, you'reÂ right!

In math, you can't take the square root of a negative number and get a real number result.Â

SoÂ 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 Â > or = to 0

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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 partiesÂ and see if it helps clarify what is happening.Â

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Stay tuned forÂ moreÂ math tips from ATT!

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