Domain and Range of a Function

Domain and Range of a Relation

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Key Questions

• Domain is the set of independent values (x) while Range is the set of dependent values (y).

Think that Domain and Range are like a picture frame for your graph. The Domain is the left and right part of the frame, while the Range is the up and down part of the frame.

To find Domain and Range, you need to graph your function. Use a graphing calculator or a table of values to create your graph. Then, analysis your graph.

To get Domain: find the left most value. If you graph goes on forever, use $- \infty$. Then, find the right most value. Again, you can use $+ \infty$ if your graph goes on forever. Make sure you need all values in between your two values.

There are two ways to write Domain: set and interval.

Set notation: D = {x| ( left value ) < x < ( right value )}.
Note, use $\le$ if you want to include any values.
Also note, if you have $D = \left\{x | - \infty < x < + \infty\right\}$ you can just use $D = \mathbb{R}$.

Interval notation: D = ( left value, right value).
Note, if you want to include any value, use a square bracket instead. So, you can get D = [left value, right value]

Now, repeat this whole process for Range. Instead of left to right, you go bottom to top.
For set notation, it looks like this: R = {y| ( bottom value ) < y < ( top value )}
For interval notation, it looks like this R = ( bottom value, top value)]

I hope this helps.

• The domain is defined as: all the numbers that you are allowed to input in the function.
In order to make the function meaningful, the output of the function must be real number .

For example,
$f \left(x\right) = 2 x + 1$

When $x = 1$,
$f \left(1\right) = 2 \cdot \left(1\right) + 1 = 3$
When $x = 2$,
$f \left(x\right) = 2 \cdot \left(2\right) + 1 = 5$
......
You can fill in every value for $x$ that you like, there is no limitation.
Therefore, the domain of the function is
"all the real numbers"

However, there are some special cases,
$f \left(x\right) = \sqrt{x}$
if $x < 0$, the output would be an imaginary number. When we're talking about function and domains, we're mostly talking about real numbers. You're therefore not allowed to input a negative number for x.
Therefore, the domain of this function is
$x \ge 0$, or in words: "all the positive numbers including zero"

$f \left(x\right) = \sqrt{x - 3}$
Inside the square root, it is again not allowed to input a negative number. So the following inequality must hold:
$x - 3 \ge 0$
$x \ge 3$ or in words "all the numbers that are greater than or equal to 3"

$f \left(x\right) = \frac{1}{x + 1}$
since the denominator can't be zero
$x \ne - 1$.

• None of the two, actually. However, the domain is related to the independent variable, as it is the set of all the "permitted" values for the independent variable to assume. In particular, given a function, you must be sure that:

1. If there is a fraction, the denominator can't be zero;
2. If there is an even root, its argument must be positive or zero;
3. If there is a logarithm, its argument must be strictly positive.

These requests can of course be combined, as for example in $\setminus \sqrt{\setminus \log \left(\setminus \frac{x}{x + 1}\right)}$.

Once you find out which values of $x$ are permitted for the expression to have sense, you take the set of all those values, and obtain the domain of the function.

• There are so many different kinds of functions, but domain and range are important parts of your study of functions.

Let me give you some examples of polynomial functions:
y = 3x + 1, y = ${x}^{2} + 3 x + 2$, and y = ${x}^{3}$. Do you notice that each one of those functions has powers of x that are Whole numbers? Stick with those, and you will have a polynomial.

All polynomials have a domain of "All Real Numbers". In interval notation, we write: $\left(- \setminus \infty , \setminus \infty\right)$. On the horizontal number line, that covers all numbers from left to right (your x-axis).

Polynomials with ODD degree (highest power of x) stretch their way from low to high through all real numbers in the vertical direction. This means that their Range is "All Real Numbers" again: $\left(- \setminus \infty , \setminus \infty\right)$. Once these functions get going in those directions, you will never see the end of them! We call this their "End behavior".

Polynomials with EVEN degree must have either a maximum or minimum value. If the graph has a minimum value, then its y-values (Range) stretch from that number, up to $\setminus \infty$. We write that Range as $\left[\min , \setminus \infty\right)$. Look at the graph shown below, it has a minimum (vertex) at (2,-4). Its Range would be $\left[- 4 , \setminus \infty\right)$.

Notice that whenever we use the $\setminus \infty$ symbols, we use a round ( or ). That means that we can not include a numeric value for the infinities. When we use the square [ or ], it refers to an actual value that is included in the function.

Your study of domain and range has just begun, and will include a wide variety of functions besides polynomials. When you discover a new function that behaves differently, carefully analyze its input and output values. You are on the way to great things!

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