Section 1-5 - Inverses

The inverse of a function is defined as f^-1(x).

2 functions; F and G, are inverses iff (f(g(x)) = g(f(x))
In other words, when the input is the output, where there is no net change.

To find the inverse of a function:
Example:
f(x) = 2x+5

f^-1(x) = ?

You would perform the following:
the original function is:
f(x) = 2x+5
and since f(x) is equal to y...
y= 2x+5
then you switch x and y...
x= 2y+5
and then you solve for y...
x-5 = 2y +5 -5
x-5= 2y
(x-5) /2 = 2y /2
y= (x-5) /2
and so:
f^-1(x) = (x-5) /2

to validate, you would show that f(f^-1(x)) = f^-1(f(x))

so;

f(f^-1(x)) = f((x-5) /2)
= ((2x+5)-5) /2
= 2x /2
= x

f^-1(f(x)) = f^-1(2x+5)
= 2((x-5) /2) +5
= x

so the inverse is correct because both equations come out to x

One-to-one means that for every y value, there is only one x value

a function f is one to one iff f(a) = f(b) implies a=b

ex:
f(x) = 2x+5
f(a) =2a+5 f(b) =2b+5
-5 -5
f(a) =2a f(b)= 2b
/2 /2
a=2=b
a=b

so the function is one to one

one-to-one functions must pass the horizontal line test, which must be true to have an inverse that is a real function.