INVERSES.-
1-To find the inverse f^-1(x) of a function f(x) we simply change the x and y of the function and the input will now become the output and the output will become the input.
FOR EXAMPLE:
f(x)= 3x-9
y=3x-9--------switching------------x=3y-9
- Likewise, graphically, the domain of f(x) will become the range of f^-1(x) and the range of f(x) will become the domain of f^-1(x).
-The points of both graphs switch with one another because the graph of f(x) and its inverse f^-1(x) are equally divided by the y=x line, meaning that they are both symmetrical about the y=x line.
FOR EXAMPLE:
f(x)= x^3 + 2
In this graph the point (0,2) lies on f(x), so its inverse f^-1(x) (here its represented by g(x)) will have the point (2,0). Notice the x switches with the y.
2.- To know if a function has an existing inverse (meaning that the inverse is also a function) we use the horizontal line test to verify. If a function does have an existing inverse then we say the function is one to one.
FOR EXAMPLE:
By using the Vertical Line Test we can prove that f(x)=x^2 is a function because it touches the graph just once!
HOWEVERIf we use the Horizontal Line Test on
f(x)=x^2 again, we can see that it doesn't have an existing inverse because it touches the graph more than once!
The inverse of f(x)=x^2 looks like this :
is the x=y^2
Notice that if we do the Vertical Line Test on the inverse function it'll touch it twice so it's not a function!
SO, AS MISS HWANG SAID, THE HORIZONTAL LINE TEST IS DONE IN THE ORIGINAL FUNCTION, THIS CASE
F(X)=X^2 SO WE ARE ABLE TO KNOW IF IT'S GOING TO HAVE AN EXISTING INVERSE BEFORE WE ACTUALY GRAPH THE INVERSE.
In this way , F(X)=X^2 is not a ONE TO ONE function.
LOGARITHMS.- :-(
OK logs are hard for me to quickly understand! But this is what I understand so far:
3.- The inverse of an exponential function f(x)= 2^x , is the logarithmic function of x, f^-1(x)=log2x because if we recall in order to find the inverse of a function we change the y with the x in the function like this:
f(x)=2^x
y=2^x
x=2^y------------------------ When we exchange the x and y , it looks like this being
the inverse!
So, if the inverse of it is f^-1(x)= log2x, it has to look like that also.
f^-1(x)= log2x
y=log2x Remember y is the exponent, 2 is the
base and x is the answer.
2^y=x------------ Yeah it does look like the one before!!!!!
THEREFORE, log2x is the
inverse of f(x) =2^x
4.- From the topic of logarithms I also know that there is this number "e" which is equals to 2.718281....and that its inverse is the natural logarithm ln.
HOWEVER::::::
I don't really understand how ln can be its inverse. First of all ...what is ln besides being the "natural logarithm". Where did it come from???...HOw Can It Be the inverse of 2.718281??
Can someone give me an example on how to solve a problem with ln AND e ?
YEAH EVERYTIME THERE IS ln AND THE THE NUMBER e INVOLVED IN A PROBLEM I GET FRUSTRATED!! :(
i have an A!!!! just checked my grade
ReplyDeleteWell for one thing natural log or ln is defined as log to the base of e.
ReplyDeleteExample: e^x = 4
Step 1: Get rid of the e by taking the ln of both sides of the equation. Its kind of like multiplying by ln but not at the same time since ln is a exponent.
ln(e^x) = ln(4)
Step 2: The ln and e cancels out and you are left with
x = ln4
wow too much info.
ReplyDeletei knowww i always write like a will ,,my frieds say that!!! when i come to see i have s lot written down ,hey but thats what i understood! and what i didn't understand of course!
ReplyDeleteThe natural log is the inverse of an exponential function with an irrational number(e)for example in te equation e^x=8 to find x you would use its inverse which is ln thus making it ln(e^x)=ln(8). So the solution would be x=ln(8)
ReplyDeleteNatural log is the same as log base of e
Well one of the few equations i understood was
ReplyDeleteIN y= 2t + 4 the natural log base is e
so you go diagonally of the
equal sign and the base will
be the base to whatever is on
the other side of the equal
sign that will equal to y
Y=e^2t+4