# Domain and Range Calculator

This calculator lets you explore the domain and range examples discussed on the previous page, Domain and Range of a Function.

As a quick refresher, recall that the domain is the set of all possible x-values which will make the function "work", and will output real y-values. Additionally, when calculating the domain be aware that the denominator of a fraction cannot be zero and the number under square root must be positive for the function to be correctly calculated.

In this interactive calculator, you can change the domain and see the effect on the range of several different trigonometric functions.

**NOTE:** We are dealing with **real numbers only** in this work.

### Things to Do

In this calculator, you start with a predefined function that has been drawn for you. You can:

- Use the slider below the curve to change the
**domain**of the function. - Observe the resulting
**range**, indicated by the height of the green rectangle, and of the pink arrow (which is solid if the range has defined limits, and dashed if the range goes off to infinity in either direction) - The range arrow is solid when the range has fixed upper and lower bounds, but is dashed when the range goes off to infnity in either direction
- Now try different functions from the "Choose function" pull-down menu at the top of the domain and range calculator.

Choose function:

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## Example explanation

The function:

**NOTE:** See Domain and Range of a Function for the original example functions used in the above calculator.

## Further Examples

Here are some more examples of domain and range.

### Example 1: Exponential Function

Consider the function `f(x) = 2^x`. You can substitute any value of `x` and will get a real value. That value is never 0, and never less than 0.

The **domain** of this function is "all possible `x`-values", which we can write as `(-oo,oo),` or `{x in RR\ |\ -oo < x < oo }.`

The **range** of this function is "all values of `y` such that `y>0`", otherwise written as `(0,oo),` or `{y in RR\ |\ y>0 }.` We don't include `0` because the curve never quite touches the `x`-axis.

See more exponential functions.

Graph of `y=2^x`, an exponential function.

### Example 2: Cubic function

Now let's consider the function `f(t) = 2t^3-6t^2+5t+1.`

The **domain** of this function is "all possible `t`-values". We could write this as `(-oo,oo)`.

The **range** of this function is "all values of `f(t)`".

There are no restrictions on the possible `t`-values, and the resulting `f(t)`-values range from `-oo` to `oo`, which we could write as `(-oo,oo)`.

Graph of a cubic function, `y=2x^3-6x^2+5x+1`.

### Example 3: Fractional function

The function `g(s) = (5s-4)/(s+3)` is known as a hyperbola. It has one vertical and one horizontal **asymptote** (a line which the curve gets closer to, but never touches).

The **domain** of this function is "all possible `s`-values, `s!=-3`", which we can write as `(-oo,-3) uu (-3,oo).` This means "the union of the sets minus infinity to minus 3, and minus 3 to infinity".

The **range** of this function is "all values of `g` such that `g!=5`", otherwise written as `(-oo,5) uu (5,oo).` We don't include `0` because the curve never quite touches the `s`-axis.

Graph of `y=(5x-4)/(x+3)`, a fraction.