math 116

What is the difference between domain and range?

The domain of a function is the set of all values for the independent variable (in the following example, that's the x) for which the function is defined. That is, if you have a function such as

f(x) = 1/x

Then this function is undefined at x=0, so the domain would be all real numbers except 0.

As another example, suppose you had the function

f(x) = sqrt(x).

We know that the the square root of a negative number is not a real number, so this function isn't defined when x is negative. So, the domain is all real numbers greater than or equal to 0.

The range, on the other hand is the set of all values that the function can take, given the domain. So, for the first example, the range is all real numbers. That is, any real number can be obtained from this function if I plug in the right value from the domain.

For the second example, the square root of a number is always positive, so the range is all real numbers greater or equal to 0.

The strategy for determining the domain and the range is this:

1. Start with the domain. See if there are any numbers that need to be excluded. For the most part, you're looking for x's in the denominator of a fraction or x's under a square root.

2. Once you figure out the domain, then try to reason out what values the function can take on given the domain that you just figured out.

• Describe a real-life situation that could be modeled by a function.

The conversion from fahrenheit to celsius is given by a function (it's even linear). f(C) = 5/9 F - 32

Describe the values for x that may not be appropriate values even when they are defined by your classmates' function.

The denominator of a fraction cannot be 0, so any time x occurs in the denominator, we must be wary of that. Furthermore, where there is a square root symbol, you cannot take the square root of a negative number, so we must be careful for that as well.

A function could, for example, indicate the amount of bone strength (y) in a living body over time in years(x). It would not make sense to look at negative years, because the person would not yet be born. Likewise, looking beyond 100 years might not make sense, as many people do not live to be 100.

These are excellent examples of values of x that would be inappropriate as well

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