00:01
If f of x is an exponential function where f at negative 2 is 8 and f at 3 .5 is 20, then find the value of f at 4 to the nearest 100th.
00:19
So if f is an exponential function, then we can say that f is of the form c times exponential of a x.
00:36
For c and a real numbers with c different from zero.
00:44
That's because if c equals zero, f will be identically null function.
00:50
So it has no sense.
00:53
It's not true because f at negative 2 is 8 and at 3 .5 is 20, so it's not a zero function.
00:58
So c got to be different from 0.
01:03
And also, a is not zero because if a is 0, then we will have c because e to the zero for any x will be one and the function will be a constant function.
01:19
That's not true because at negative 2 we have a value and at 3 .5 we have another one.
01:26
So both a and c are not zero.
01:30
So we got to find c and a.
01:33
And for that we use the two conditions that f at negative 2 is 8 and f at 3 .5 is 20.
01:40
So we have 8 equal.
01:46
F at negative 2 and that is c exponential of negative 2a and the other condition is 3 .5 sorry 20 is equal let's say here and f at 3 .5 and that is c exponential of 3 .5 a so from the first equation where we have no decimals for the moment.
02:33
So we have from 8 equal c e to the negative 2a, which is c is the same as c over exponential of 2a.
02:53
We get that c is 8 exponential of 2a.
03:10
That is we write c in terms of a and and so we need to calculate a, and that will be all.
03:23
And the other thing we can do is find c in the other equation, 20 equal c to the 3 .5a implies that c is 20 over exponential of 3 .5a.
03:57
And because c is also equal to 8 exponential of 2a, and then using equation asterisk, let's say this one here, we get that 20 over exponential of 3 .5a get to be equal to 8 e to the 2a.
04:44
Okay? then we pass 8 divided into the left and exponential of 3 .5 to the right.
04:55
So we get 20 over 8 equal e to the 3 .5a times e to the 2a.
05:08
That is, 20 over 8 can be simplified by divided by 4...