00:01
The question i have is asking me to determine the properties of three different functions.
00:06
I have the inverse function, 1 over x, and i have ln of x, and i have e of x.
00:12
So we are going to start with the first one, the inverse function, f of x is equal to 1 over x.
00:19
The first thing i need to determine is the domain.
00:22
I know that the fraction cannot have a denominator equal to 0.
00:25
So the domain for this function is going to be all the numbers, all the possible values of x, except x equals to 0.
00:33
I cannot have x equals to 0 into the denominator.
00:36
For the range, this is a fraction, and the fraction will only be 0 if the numerator is 0.
00:43
The numerator here is always 1, so that means my function will never be equal to 0.
00:56
Now, the third thing that i need to determine is the 0s or the possible 0s for this function.
01:03
When does the function equal to 0? the answer is never because we just mentioned in the range that the y values will not be zero because my numerator is always going to be 1.
01:16
The first thing that i need to talk about is when is f of x going to be positive? i have a function of 1 over x.
01:25
If i replace x by a positive number, then my fraction will be net positive.
01:30
If i replace x by a negative number, my fraction will be negative.
01:34
So f of x is going to be positive when x itself is positive, and then f of x is going to be negative, when x is negative.
01:43
Now, about the end behavior, if x becomes very big, we notice that the fraction will get smaller and smaller.
01:53
So that means when x goes to infinity, when x becomes very big, the function approaches zero.
01:59
So we're going to say that f of x is close to zero or has an asymptote at zero okay, and this happens from both sides from the negative side as well as the positive side now i have to put all of this information together in order to graph the function this is my x and y axis i have a positive side of the function and also a negative side my function will never be be zero so that means the lines i will draw will never touch the x or the y axis okay if x is equal to one my function will be equal to one over one which is also one and if x is equal to negative one my function will become equal to negative one as well and i could choose a third point which is for example this is equal to two so if i had a place x by 2, my function is equal to 1 over 2, which is 1 half.
03:43
Okay? so i'm going to try to place these points.
03:47
So this is going to be 1 and 1.
03:49
That's my first point.
03:50
I got negative 1, and then negative 1, this is going to be my second.
03:54
And then when f is equal to 2, then my function is going to be 1⁄2.
03:59
My graph is going to look something like.
04:03
Okay? we want to be careful that towards the end or the one.
04:16
So the graph is going to look something like this.
04:20
Now, let's talk about the second function, which is ln of x.
04:26
So what is the domain? what are the possible values that ln of x or x could take in this particular function? i know for a logarithmic function, x cannot be negative because the logarithmic function is not defined for a negative number.
04:43
That means my domain is going to be x is bigger than zero.
04:47
For the range, this number, this function could have all different values.
04:52
So that means the range is going to be all the numbers.
05:02
The zeros, i know that ln of 1 is equal to 0.
05:07
So that means it will have a 0 when x is equal to 1.
05:26
My function is going to be positive when x is bigger than 1...