Domain and Range
The domain of a function f(x) is the set of all values of x for which f(x)is defined.
The range of a function f(x) is the set of all values of f(x), where x is in the domain of f.
For odd numbered radicals both the domain and range span all real number.
For even numbered radical functions, the term inside the radical must be at or above zero, otherwise it is undefined. This means that only the x values that make the term inside an even numbered radical positive are defined and in the domain.
Example: f(x)=√ (x-3)+4
Since (x-3) is inside the radical, the domain lies on all the points where x makes (x-3) greater than or equal to zero.
x-3 ≥ 0
x ≥ 3
This occurs at any point at or above three, so the domain of the function is [3,∞).
The range of the function is then all the points of the y-axis that x hits for the given values of the domain.
We start at the point x=3 and plug it into the equation.
f(x)=√ (3-3)+4=√ (0)+4=4
It is then easy to see that by plugging in any number larger than 3 for x will result in an f(x) larger than 4, so the smallest number in the range is 4.
It is not easy to see that the range will go to infinity, but it will. For any number above 4 that is set as the answer to the above function, there will also be an x to define it. Therefore the range is [4,∞).
Example: y=-√ (-x-3)+4
This example is a little confusing because of the inserted negatives before the radical and before the independent variable.
For the domain we remember that (3-x) has to be greater than zero.
This means that -x has to be greater than or equal to negative three.
-x≥3
When you multiply both sides by negative one to solve for x, you also have to flipped the inequality.
x≤-3
Therefore, the domain of the function is (-∞,-3]. Notice how putting a negative in front of the x flips the domain upside down.
We then find the range the same way we did last time. Again, start where x=-3.
y=-√ [-(-3)-3]+4=-√ (3-3)+4=-√ (0)+4=4
This time it seems that when we plug more numbers from the domain into the function, the function continues to decrease. This is not because the domain lies in the negative numbers, but because of the negative sign in front of the radical. This function also goes to infinity, only negative infinity this time, since the function is decreasing. Therefore, the range of this function is (-∞,4].
The range of a function f(x) is the set of all values of f(x), where x is in the domain of f.
For odd numbered radicals both the domain and range span all real number.
For even numbered radical functions, the term inside the radical must be at or above zero, otherwise it is undefined. This means that only the x values that make the term inside an even numbered radical positive are defined and in the domain.
Example: f(x)=√ (x-3)+4
Since (x-3) is inside the radical, the domain lies on all the points where x makes (x-3) greater than or equal to zero.
x-3 ≥ 0
x ≥ 3
This occurs at any point at or above three, so the domain of the function is [3,∞).
The range of the function is then all the points of the y-axis that x hits for the given values of the domain.
We start at the point x=3 and plug it into the equation.
f(x)=√ (3-3)+4=√ (0)+4=4
It is then easy to see that by plugging in any number larger than 3 for x will result in an f(x) larger than 4, so the smallest number in the range is 4.
It is not easy to see that the range will go to infinity, but it will. For any number above 4 that is set as the answer to the above function, there will also be an x to define it. Therefore the range is [4,∞).
Example: y=-√ (-x-3)+4
This example is a little confusing because of the inserted negatives before the radical and before the independent variable.
For the domain we remember that (3-x) has to be greater than zero.
This means that -x has to be greater than or equal to negative three.
-x≥3
When you multiply both sides by negative one to solve for x, you also have to flipped the inequality.
x≤-3
Therefore, the domain of the function is (-∞,-3]. Notice how putting a negative in front of the x flips the domain upside down.
We then find the range the same way we did last time. Again, start where x=-3.
y=-√ [-(-3)-3]+4=-√ (3-3)+4=-√ (0)+4=4
This time it seems that when we plug more numbers from the domain into the function, the function continues to decrease. This is not because the domain lies in the negative numbers, but because of the negative sign in front of the radical. This function also goes to infinity, only negative infinity this time, since the function is decreasing. Therefore, the range of this function is (-∞,4].
