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## Algebra I (2018 edition)

### Course: Algebra I (2018 edition)>Unit 6

Lesson 5: Introduction to the domain and range of a function

# 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?
• The arrows simply mean that the function goes on forever.
• How do you find the domain of a parabola? Do you use the same process?
• 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.
• So essentially we can interpret this as Domain being represented along the X-axis and Range along the Y-axis?
• -2<x<5 how can i write the inequalities?
• 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.
• 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
• 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.
• 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)?
• Yes, but you should always right the range in numeric order: (0,∞).
• What is a function?

I keep confusing myself on what it is...
I know domain is x and range is y
• 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
• how high is up
• what do I do if there are 2 points on one side of the domain and not a closed or open circle on the other side?
• How do you graph this domain? 0 is less than or equal to x, which is less than or equal to 20?

## 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.