Determining domain and range
Domain: x values, inputs of a function, numbers that you are allowed to put in a function
Range: y values, outputs of a function
Range: y values, outputs of a function
If the domain is not all real numbers, then it must be restricted. We can restrict our domain using interval notation:
() parenthesis = not inclusive = open dot = < or >
[] brackets = inclusive = closed dot = < or >
u = union symbol (no overlap)
n = overlap
Numbers included are part of the possible domain while excluded numbers are not part of the possible domain.
Look at the figure above. We used a bracket before the 0 to show that 0 is included in the domain, and the bracket behind the 20 shows that 20 is also included in the domain of the function. We put a comma between the two numbers.
() parenthesis = not inclusive = open dot = < or >
[] brackets = inclusive = closed dot = < or >
u = union symbol (no overlap)
n = overlap
Numbers included are part of the possible domain while excluded numbers are not part of the possible domain.
Look at the figure above. We used a bracket before the 0 to show that 0 is included in the domain, and the bracket behind the 20 shows that 20 is also included in the domain of the function. We put a comma between the two numbers.
Now let's put interval notation in context. The figure above shows how we can use a number line to help us write the domain of a function in interval notation. We will read this graph from left to right. Notice the lines and dots.
0 has a closed dot so... [0
4 has an open dot so... 4)
4 will come after 0 because it is farther right on the number line... [0,4)
We do not need a union symbol because there are no breaks in the domain. Our final domain for this function is [4,0)
- closed dot = included in domain = use brackets []
- open dot = not included in domain = use parenthesis ()
0 has a closed dot so... [0
4 has an open dot so... 4)
4 will come after 0 because it is farther right on the number line... [0,4)
We do not need a union symbol because there are no breaks in the domain. Our final domain for this function is [4,0)
Examine the figure above. We have a closed dot at 2 and an arrow going on negatively from 2 forever. If a graph has an arrow, it means infinity or negative infinity. Here we have an arrow going onto negative infinity. Remember that infinity and negative infinity are NOT OBTAINABLE! Therefore, both positive and negative infinity will ALWAYS use parenthesis ().
Our final domain for this function is (-infinity, 2] because negative infinity cannot be obtained, and 2 is obtained because of the closed dot.
Our final domain for this function is (-infinity, 2] because negative infinity cannot be obtained, and 2 is obtained because of the closed dot.
We will use a union symbol for the example above because there is a break in the domain. We have a closed dot at 2 and an open dot at 3.
2 has a closed dot... [2
3 has an open dot... (3
However, the break in the domain means that we must use our union symbol "u" between our 2 domain pieces. Here is our final domain:
(-infinity, 2] u (3, infinity)
Remember that infinity is always noninclusive and the "u" means that both pieces of the graph are part of the possible domain.
2 has a closed dot... [2
3 has an open dot... (3
However, the break in the domain means that we must use our union symbol "u" between our 2 domain pieces. Here is our final domain:
(-infinity, 2] u (3, infinity)
Remember that infinity is always noninclusive and the "u" means that both pieces of the graph are part of the possible domain.
Assigning domain and range
We can show domain and range by using a colon and brackets:
Domain:{}
Range:{}
If the domain and range are all real numbers, like in the example below, we can use the double-backed-R to show that all real numbers are included in the domain and range: IR.
Domain:{}
Range:{}
If the domain and range are all real numbers, like in the example below, we can use the double-backed-R to show that all real numbers are included in the domain and range: IR.
It is very easy to find the domain and range of a cluster of points. The domain will include all of the x values, and the range will be all of the y values (it is helpful to write down the coordinates of all the points).
Follow the format, it is customary not to repeat numbers and to put numbers in numerical order.
Domain: {-3, -1, 0, 2}
Range: {-2.5, 0, 1, 3}
The line- and function- to the left has a domain and range of all real numbers because, as the arrows indicate, the graph goes on forever both negatively and positively.
The domain and range are all real numbers because, at some point, the x and y values will be every real number.
Domain: {IR}
Range: {IR}
We could also use interval notation to assign our domain and range:
The domain and range are all real numbers because, at some point, the x and y values will be every real number.
Domain: {IR}
Range: {IR}
We could also use interval notation to assign our domain and range:
- Domain (-infinity, infinity)
- Range (-infinity, infinity)
The square root function to the right does not have a domain or range of all real numbers. Remember that a domain and range indicate what x and y values, respectively, can exist for the equation.
Let's try to determine the domain first. Look at the x values. The function begins at 1, so our possible domain values also begin at 1, and the values continue positively after 1.
Therefore, our domain is: Domain: {x>1}.
Or, we could assign our domain using interval notation: [1,infinity).
Our range, or y values, begin at 2 and continue positively after 2.
Therefore, our range is: Range: {y>2}.
Again, we could use interval notation to assign our range: [2,infinity)
We can check our answer by looking at the graph. According to the domain and range values we determined, (0,0) could not be a part of the range for this function. The graph agrees with this conclusion. If we find (0,0), the square root function is undetermined at that point and does not appear to exist, so we now have evidence that our domain and range are correct.
Let's try to determine the domain first. Look at the x values. The function begins at 1, so our possible domain values also begin at 1, and the values continue positively after 1.
Therefore, our domain is: Domain: {x>1}.
Or, we could assign our domain using interval notation: [1,infinity).
Our range, or y values, begin at 2 and continue positively after 2.
Therefore, our range is: Range: {y>2}.
Again, we could use interval notation to assign our range: [2,infinity)
We can check our answer by looking at the graph. According to the domain and range values we determined, (0,0) could not be a part of the range for this function. The graph agrees with this conclusion. If we find (0,0), the square root function is undetermined at that point and does not appear to exist, so we now have evidence that our domain and range are correct.
The parabola's x values will eventually be every real number.
The range is y>2 because y values for this function only exist after y>2. y = 4 would be included in the domain, but y = -4 would not be included.
Domain: {IR}
Range: {y>2}
Domain in interval notation: (-infinity, infinity)
Range in interval notation: [2,infinity)
The range is y>2 because y values for this function only exist after y>2. y = 4 would be included in the domain, but y = -4 would not be included.
Domain: {IR}
Range: {y>2}
Domain in interval notation: (-infinity, infinity)
Range in interval notation: [2,infinity)