I know the domain is [-6,infinity) and that the range is [0,infinity), but how do they get the range? I know how to get the domain but not the range. Can someone teach me???
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For the range, you have to think about the function. what is the minimum value of sqrt[a]?
sqrt[0] = 0
sqrt[-1] = undefined
sqrt[1] = 1 {generally the sqrt function is defined as the positive root}
If you think about what the graph looks like, this is the top half of a parabola opening to the right.
Then as x increases, sqrt[x] increases
sqrt[infinity] = infinity {well kind of, anyway}
So then the range is from the min of 0 to infinity.
Basically on the range, you have to compare the function you are given, and apply the domain to it to make sure that doesn't limit the range any.
There isn't really one way to find domains and ranges. There are just a group of things that we typically look for.
e^x domain is all real #
Range is ???
is there any possible exponent you can have on a positive number that will make the number = 0?
No
So 0 cannot be in the range.
But what is the limit as x --> -infinity? 0
So y approaches 0., but is never 0. giving us the left side boundary (0,
As x approaches +infinity, e^x will also approach +infinity.
Are there any values missing in the range? no. So the range is (0, +infinity)
How about x (x+2)/(x+2)
This is y = x everywhere but x = -2.
At x = -2, the function is undefined, so -2 is not in the domain.
Since y = x everywhere but at x = -2, then y = -2 cannot be part of the range either.
So the range is the reals without y = -2
Conceptually these usually aren't too hard. There is just a lot of possibilities to cover.
Start with rationals. Anything that makes a denominator = 0 is a red flag.
radicals, especially even ones. sqrt, 4thrt, 6thrt... etc.
You cannot take even roots of negative numbers in the real numbers. So that is a red flag.
Then there are functions that just by their normal characteristics have certain boundaries.
a^x will never be 0 for any positive a.
sin[x] inherently has a range from -1 to 1
Be careful of horizontal asymptotes. Some graphs can cross them.
sqrt[0] = 0
sqrt[-1] = undefined
sqrt[1] = 1 {generally the sqrt function is defined as the positive root}
If you think about what the graph looks like, this is the top half of a parabola opening to the right.
Then as x increases, sqrt[x] increases
sqrt[infinity] = infinity {well kind of, anyway}
So then the range is from the min of 0 to infinity.
Basically on the range, you have to compare the function you are given, and apply the domain to it to make sure that doesn't limit the range any.
There isn't really one way to find domains and ranges. There are just a group of things that we typically look for.
e^x domain is all real #
Range is ???
is there any possible exponent you can have on a positive number that will make the number = 0?
No
So 0 cannot be in the range.
But what is the limit as x --> -infinity? 0
So y approaches 0., but is never 0. giving us the left side boundary (0,
As x approaches +infinity, e^x will also approach +infinity.
Are there any values missing in the range? no. So the range is (0, +infinity)
How about x (x+2)/(x+2)
This is y = x everywhere but x = -2.
At x = -2, the function is undefined, so -2 is not in the domain.
Since y = x everywhere but at x = -2, then y = -2 cannot be part of the range either.
So the range is the reals without y = -2
Conceptually these usually aren't too hard. There is just a lot of possibilities to cover.
Start with rationals. Anything that makes a denominator = 0 is a red flag.
radicals, especially even ones. sqrt, 4thrt, 6thrt... etc.
You cannot take even roots of negative numbers in the real numbers. So that is a red flag.
Then there are functions that just by their normal characteristics have certain boundaries.
a^x will never be 0 for any positive a.
sin[x] inherently has a range from -1 to 1
Be careful of horizontal asymptotes. Some graphs can cross them.