Main content

## Introduction to the domain and range of a function

# Worked example: domain and range from graph

CCSS.Math:

## 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.