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For y=12, there are two possible x’s. x=-4, and x=4.

(-4,12)

(4,12)

However, for each x there is only one possible y, so y=x2-4 is a function.

Only one-to-one functions have inverses. Not all functions have inverses because all functions are not one-to-one functions.

Definition of a one-to-one function: A function is a one-to-one if no two different elements in the domain have the same element in the range. The definition of a one-to-one function can be written algebraically as follows: A function f(x) is one-to-one

if x1 is not equal to x2 (x1 and x2any elements of the domain)

then f(x1) is not equal to f(x2).

In other words, for any two ordered pairs (x1,y1) and (x2, y2) ,

where y1 = f(x1) and y2 = f(x2),

Then if x1 ≠ x2, then y1 ≠ y2.

Similarly, if f(x1)= f(x2), then it must be that x1 = x2.

Just as we had a vertical line test to test if a graph represents a function, there is a horizontal line test to test if a function is 1-to-1.

Horizontal Line Test Theorem

If every horizontal line intersects the graph of a function f in at most one point, then f is 1-to-1.

Below is the graph of y=x2-4

Does not pass

Horizontal Line

Test

Therefore, this function is not 1-to-1.

What would the inverse fuction of y = x2 -4 be?

Solve for x.

y + 4 = x2

A function is 1-to-1 over a certain interval

only if it is constantly decreasing or constantly increasing over that interval. y=x2-4 is 1-to-1 over the intervals (-∞,0) and (0, ∞)

Which one do we choose?

We need to have a specific value