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### Course: College Algebra > Unit 3

Lesson 3: Domain and range of a function- Intervals and interval notation
- What is the domain of a function?
- What is the range of a function?
- Worked example: domain and range from graph
- Domain and range from graph
- Determining whether values are in domain of function
- Examples finding the domain of functions
- Determine the domain of functions

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# Worked example: domain and range from graph

Finding the domain and the range of a function that is given graphically. Created by Sal Khan.

## Want to join the conversation?

- What would I write if the function has arrows at the end of the line on both sides?(31 votes)
- The arrows simply mean that the function goes on forever.(42 votes)

- How do you find the domain of a parabola? Do you use the same process?(16 votes)
- A parabola should have a domain of all real numbers unless it is cut off and limited. Both the left side and the right side normally have arrows which mean it will go on forever to the left and forever to the right.(11 votes)

- So essentially we can interpret this as Domain being represented along the X-axis and Range along the Y-axis?(10 votes)
- yes Domain is the X axis and range is the Y axis(4 votes)

- I'm confused on what signs to use (greater than equal to, less than equal to, etc) I know that you use the greater than equal to and less than equal to, when it's included, but how do you know what sign to use when graphing? How do you know which way the graph is going? I'm not sure if I am making sense(9 votes)
- The "equal" part of the inequalities matches the line or curve of the function, so it would be solid just as if the inequality were not there. Without the "equal" part of the inequality, the line or curve does not count, so we draw it as a dashed line rather than a solid line

The < or > has to do with the shading of the graph, if it is >, shading is above the line, and < shading is below. The exception is a vertical line (x = #) where there is no above and below, so it changes to the left (<) or to the right (>)..

So lets say you have an equation y > 2x + 3 and you have graphed it and shaded. If you try points such as (0,0) and substitute in for x and y, you get 0 > 3 which is a false statement, and if you did it right, shading would not go through this point. If point is (1,5) you can do the same thing, 5 > 5, but this would be right on the line, so the line would have to be dashed because this statement is not true either. One more point (0,6) would give 6>3 which is a true statement, and shading should include this point.

Does this answer your question?(7 votes)

- -2<x<5 how can i write the inequalities?(5 votes)
- You would write your inequality in interval notation as:

(-2, 5)

The parentheses tell you that the inequalities do not include the end values of -2 and 5.

If the inequality is: -2≤x≤5, then the interval notation is:

[-2, 5]

The square brackets tells you that the end values are included in the interval.

If you have an inequality like: -2≤x<5, then the interval notation is:

[-2, 5)

A square bracket is on the -2 because it is included in the interval. The 5 gets a parentheses because it is not in the interval.

Hope this helps.(14 votes)

- What is a function?

I keep confusing myself on what it is...

I know domain is x and range is y(5 votes)- A function is a relation where every domain (x) value maps to only one range (y) value.

If you have the points (2, -3), (4, 6), (-1, 8), and (3, 7), that relation would be a function because there is only one y-value for each x. X-values don't repeat.

If you have the points (2, -3), (4, 6), (2, 8), and (3, 7), that relation would not be a function because 2 for the x-value repeats, meaning 2 maps to more than one y-value.

Repeating x-values mean the relation is not a function.

No repeating x-values mean the relation is a function.

You might want to check out https://www.khanacademy.org/math/algebra/algebra-functions/evaluating-functions/v/what-is-a-function(7 votes)

- If we have f(x)=(1/3)^x, we can see that it approaches 0, but never touches it. Does that mean that the range is (∞,0)?(4 votes)
- Yes, but you should always right the range in numeric order: (0,∞).(7 votes)

- so for the domain you dont need the greatest point the line touches in the graph, but you need that for the range..? i need clarification please(5 votes)
- Domain is all the values of X on the graph. So, you need to look how far to the left and right the graph will go. There can be very large values for X to the right.

Range is all the values of Y on the graph. So, you look at how low and how high the graph goes.

Hope this helps.(4 votes)

- What website is Sal using for these questions?(4 votes)
- Sal is using Khan academy, but since this video was recorded a while ago, it looks different from the Khan academy that we use today.(4 votes)

## Video transcript

The function f of x is graphed. What is its domain? So the way it's graphed
right over here, we could assume that this
is the entire function definition for f of x. So for example, if
we say, well, what does f of x equal when x
is equal to negative 9? Well, we go up here. We don't see it's graphed here. It's not defined for x
equals negative 9 or x equals negative 8 and 1/2 or
x equals negative 8. It's not defined for
any of these values. It only starts getting defined
at x equals negative 6. At x equals negative 6,
f of x is equal to 5. And then it keeps
getting defined. f of x is defined for x all
the way from x equals negative 6 all the
way to x equals 7. When x equals 7, f
of x is equal to 5. You can take any x value
between negative 6, including negative
6, and positive 7, including positive
7, and you just have to see-- you
just have to move up above that number,
wherever you are, to find out what the value of
the function is at that point. So the domain of this
function definition? Well, f of x is
defined for any x that is greater than or
equal to negative 6. Or we could say negative 6
is less than or equal to x, which is less than
or equal to 7. If x satisfies this
condition right over here, the function is defined. So that's its domain. So let's check our answer. Let's do a few more of these. The function f of x is graphed. What is its domain? Well, exact similar argument. This function is not defined
for x is negative 9, negative 8, all the way down or all the way
up I should say to negative 1. At negative 1, it
starts getting defined. f of negative 1 is negative 5. So it's defined for negative
1 is less than or equal to x. And it's defined all the
way up to x equals 7, including x equals 7. So this right over
here, negative 1 is less than or equal to x
is less than or equal to 7, the function is
defined for any x that satisfies this double
inequality right over here. Let's do a few more. The function f of x is graphed. What is its range? So now, we're not
thinking about the x's for which this
function is defined. We're thinking about
the set of y values. Where do all of the
y values fall into? Well, let's see. The lowest possible y value
or the lowest possible value of f of x that we get
here looks like it's 0. The function never goes below 0. So f of x-- so 0 is less
than or equal to f of x. It does equal 0 right over
here. f of negative 4 is 0. And then the highest y
value or the highest value that f of x obtains in this
function definition is 8. f of 7 is 8. It never gets above 8, but it
does equal 8 right over here when x is equal to 7. So 0 is less than f of x, which
is less than or equal to 8. So that's its range. Let's do a few more. This is kind of fun. The function f of x is graphed. What is its domain? So once again, this function
is defined for negative 2. Negative 2 is less than or
equal to x, which is less than or equal to 5. If you give me an x anywhere
in between negative 2 and 5, I can look at this graph to see
where the function is defined. f of negative 2 is negative 4. f of negative 1 is negative 3. So on and so forth,
and I can even pick the values in
between these integers. So negative 2 is less than or
equal to x, which is less than or equal to 5.