00:01
Okay, so here we are going to study the function f of x equals x squared minus 4 over x plus 1.
00:12
Okay, the first thing that we need to do is find the domain of f.
00:18
Now what is the domain of f? the domain of f is negative infinity comma negative 1, union, negative 1, negative 1.
00:30
1 comma infinity.
00:33
That is, f is defined everywhere, but at the point x equals negative 1, which is the 0 of the denominator.
00:44
So the first thing that we can say here is vertical asymptote, x equals negative 1.
00:54
Okay.
00:56
Now, what are we going to do next? we are going to find the x and the.
01:01
And y intercepts of this function.
01:05
So the x intercept and an x intercept of this function is just what, a zero of this function.
01:14
So x intercept here we're going to have two values.
01:24
We are going to have x equals plus or minus two.
01:28
So x equals plus or minus two.
01:32
Now, y intercepts.
01:36
Okay, this one is just the value of f at x equals 0.
01:42
So y intercept.
01:50
Here we got, for x equals zero, we got negative 4.
01:55
So y equals negative 4.
01:58
Okay, perfect.
02:01
Now let's see what happens as x goes to infinity.
02:06
So we're going to find the horizontal asymptotes okay actually i'm going to write this thing here so horizontal asymptoped well when x goes to infinity f of x goes to infinity same thing when x goes to negative infinity so known perfect now let's study the behavior of or the differential behavior of f that is we are going to study when f is increasing decreasing by using the first derivative okay so maybe it's better if i write the first derivative here f prime of x is equal to what here we're going to use the quotient rule is x plus 1 squared and we're going to have 2x multiplied by x plus 1 so 2x squared plus 2x plus 2x minus x squared plus 4 so minus x squared plus 4 okay well now this one is equal to what this one is equal to x squared plus 2x plus 4 over x plus 1 squared now as we can see the numerator of this rational function here does not have any zero because the discriminant is less than zero.
03:52
So in particular, this means that f prime of x is always greater than zero.
03:57
So we can conclude that f is increasing everywhere...