Calculus > How to Find the Domain and Range of a Function

## What is a “Domain” and a “Range”?

- The
**domain**is the set of x- values that can be put into a function. In other words, it’s the set of all possible values of the independent variable. - The
**range**is the set of y- values that are output for the domain.

## Four Ways to Find the Domain and Range of a Function

## 1. Check for Known Domains/Ranges

See if you can figure out what type of function you have first (this isn’t always clear).

Many functions have an infinite set for the domain. An “infinite set” is just the set of all possible numbers. For example, you could input any number you like into the function y = x^{2}, and it will still give you an output. But what about the range? A negative number will never show for this function; a negative times a negative will always be positive. If you put, for example, -10 in, you get:

y = -10^{2} = -10 * -10 = 100.

It makes sense that the range for x^{2} is 0 > ∞.

**Certain functions have defined domains and range. **

*Linear functions*have the domain and range of all real numbers*Polynomial functions*have the domain of all reals, and the range of all positive reals.*Square (quadratic) functions*and*absolute value functions*have a domain of all real numbers and a range of y ≥ 0*Square root functions*have a domain of x ≥ ;0 and a range of y ≥ 0*Rational functions*have a domain of x ≠ 0 and a range of x ≠ 0.*Sine functions*and*cosine functions*have a domain of all real numbers and a range of -1 ≤y≥ 1.

**Tip:** Become familiar with the shapes of basic functions like sin/cosine and polynomials. That way, you’ll be able to reasonably find the domain and range of a function just by looking at the equation.

## 2. Guess and Check

**If you don’t have strong algebra skills, you may want to skip this method** and try the graph or table methods instead.

Basically, use your algebra skills to find the domain and range for a function by guessing and checking! Some general tips:

**Division by zero is not allowed**). As an example, let’s say you have the function:

f(x) = 1/(x^{2}– 9).

You can exclude any values of x (the domain) that make the denominator equal to zero.- For a domain,
**the number under a square root sign can’t be negative**. For example, you can’t find the domain for √-10, because the solution is an imaginary number. **Try putting different x-values into the function**for y to see what happens. Look for trends like: always positive, always negative, or sets of numbers that don’t work. Try putting in very large (e.g. a million), or very small (e.g. negative million) and see if those work.

**Example**: Find the domain and range of a function with algebra

Find the domain and range for:

**Domain:**

- The
*numerator*has a square root; numbers under this can’t be negative (see #2 above). So you can only have numbers for x greater than or equal to -2. - The
*denominator:*You can’t have division by zero, you can’t have -3 + 3 as this would result in zero. For example, 3^{2}– 9 = 0.

The domain for this particular function is x > -2, x ≠ 3.

**Range**

The range for this function is the set all values of f(x) excluding F(x) = 0. Here’s where your algebra skills get a workout!

**Numerator**: By looking at the function, you should immediately see that the numerator becomes 0 when x = -2:

√(2 + 2) = √0 = 0.**Denominator:**- Working with -2 still, the denominator becomes: (-2)
^{2}– 9 = 5.So f(-2) = 0/-5 = 0. - If you
**insert a few x-values**between 2 and 3 into (x^{2}– 9), you’ll see that the function approaches negative infinity. - Insert some more x-values greater than x = 3, note that the function tends toward positive infinity.
- The larger the x-values get, the smaller the function values get (but they never actually get to zero).

- Working with -2 still, the denominator becomes: (-2)

## 3. Graphing

**Graph your function **and see where your x-values and y-values lie. Most graphing calculators will help you see a function’s domain (or indicate which values might not be allowed). For example, if you graphed x^{2}, it would be clear that the domain cannot include negative numbers. If you don’t have a graphing calculator, try this free online one. **Always zoom in and zoom out of the graph to check for continuity or missing areas.**

From the above graph, you can see that the range for x

^{2}(green) and 4x

^{2}+25 (red graph) is positive; You can take a good guess at this point that it is the set of

*all*positive real numbers, based on looking at the graph.

## 4. find the domain and range of a function with a Table of Values

Make a table of values on your graphing calculator (**See**: How to make a table of values on the TI89).

Include inputs of x from -10 to 10, then some larger numbers (like one million). Use the calculator to find values of y for values of x. If the calculator tells you the values or undefined, or that the values might be reaching a limit (a number that a function approaches, but never reaches), that should help you determine the range.

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