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# Domain and Range of a Function

### Domain and Range of a Function

Functions include a range of possible input and output values, called the domain and range.

You can think of functions a bit like a machine. All machines have an input, a process and an output. Algebraically, a function performs the process that takes an input to an output.

The **domain** of a function includes all of the allowable input values.

The **range** of a function includes all of the possible output values.

The domain of the functions contains...

The range of a function contains...

Sometimes you will have to state exclusions to the domain. The two main things to look for are:
*Any value that results in* **division by zero***in the function*

Any value that results in the **square root of a negative number**

For example, what value must be excluded from the domain for $f(x) = \dfrac{3}{x + 1}$

Let's try another one! Which values must be excluded from the domain for $g(x) = \sqrt{x}$

What value must be excluded from the domain for $f(x) = \dfrac{2}{x - 1}$

An **inverse** function reverses the effect of a function.

$f(x) = x + 2$ This function that adds 2 to $x$, so the inverse of this would be subtracting 2 from $x$. The notation for the inverse function would be $f^-1(x) = x-2$

Find the inverse of $f(x) = 2x$

Graphically, an inverse function is the reflection of a function in the line $y = x$

This also means that the **domain** of a function has the same values as the **range** of its inverse.

For example, in the graph above the blue function has a domain of $x\geq0$. Notice that the range of the green function is $f^-1(x)\geq0$