4-5 Inverse Functions.

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Presentation transcript:

4-5 Inverse Functions

First, lets define something. A One to One Function – is a function where each y has a unique x. (1, 1) (2, 2), (3, 5), (4,12) is one-to-one (1, 1) (2, 2), (3, 2), (4,12) is not.

What was the graphic way to test functionality? For regular functions, you did a vertical line test. For one to one functions, they ALSO pass a horizontal line test. That is, as long as each x has its own unique y that no other x has, it is one to one. The connection will be made later.

What is an inverse function? Two functions are said to be inverses iff f(g(x)) = g(f(x)) = x. That is, the composite of a function with its own inverse is x. Now what does that mean? It means that they are functions that “negate” each other, such as and .

So, lets experiment Let and What is f(g(x))? How about g(f(x))? Find f(0), f(1), f(2) and f(3). Now find g(0), g(1), g(4) and g(9). Make a table for both functions. Do you see it yet??

So what is going on? That is an inverse – for each ordered pair (x,y) the inverse creates the ordered pair (y,x). Notation for an inverse if it is a function if it is NOT a function

The visual of an inverse Graph the two data sets on a single coordinate axis. What do you see? If functions are inverses, then they are reflections across the line y = x.

f(x) (1,2) f-1(x) (2,1)

How do I FIND an inverse? The easiest way is this: Replace every x with a y and every y with an x, then solve for y. How to test? Take the composite with its possible inverse – if you get x, then its correct.

Examples