Today in class we went over the inverse function. We first went over how to find the inverse of a single function. For example: refer to the function f(x)=2x+5

Step 1: replace the f(x) with the letter y;
y=2x+5
Step 2: swap x with y;
x=2y+5
Step 3: solve for y
(x-5)/2=y

We then learned that the definition of an inverse could be expressed as (fof^-1)(x)=x
and (f^-1of)(x)=x. Refer to the function above for this example on verifying functions as an inverse.

f(f^-1(x))=f(x-5)/2
=2(x-5)/2+5
=x-5+5
=x; because we got x as the answer, we therefore verified this function as
an inverse!
Just to double check we will also prove this function as an inverse with the second definition given above.

f^-1(f(x))=f^-1(2x+5)
=(2x+5-5)/2
=2x/2
=x; because we got x as the answer, we therefore verified this function as
an inverse!

The two examples above show how to prove that a function has an inverse algebraically. We can also prove this by graphing. The way to check if a function has an inverse is if the inverse reflects y=x. What this is saying is that if f(2, 7) is on a graph than so is its inverse of f^-1(7, 2)

We then learned that some functions do not have inverses :( These functions include...
-absoloute value
-functions that don't pass the horizontal line test

We then learned a very helpful way to test if a function has a true inverse. You can check if a function is one-to-one to verify if it has an inverse.
-One-to-one means...
if and only if (iff) f(a)=f(b), implies a=b

Examples!
f(x)=x^2
a^2=b^2
a=-b or a=b, because a could equal b OR -b, this is not a one-to-one function

f(x)= 2x+5
f(a)= 2a+5; f(b)=2b+5
2a+5=2b+5
2a=2b
a=b; because we got one solution for a... which should always be b, this is a one-to-one function, so it has an inverse!

Tonight's homework is...
Assignment 8: 1.5 problems 16, 21, 31, 43, 46, 54, 56, 73, 81, 83, 85, 88, 92, 93, 96