One-to-One Functions A function is one-to-one if no two elements of A have the same image, or f(x1)  f(x2) when x1  x2. Or, if f(x1) = f(x2), then.

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Presentation transcript:

One-to-One Functions A function is one-to-one if no two elements of A have the same image, or f(x1)  f(x2) when x1  x2. Or, if f(x1) = f(x2), then x1 = x2. Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more than once. Ex 1: Is a one-to-one function? Yes

Ex 2: Is a one-to-one function? No Ex 3: Is a one-to-one function? Yes

Inverse Functions Def: If f is a one-to-one function with domain A and range B, then it’s inverse function f-1 has domain B and range A and is defined by f-1(y) = x f(x) = y for any y in B. Ex 4: If f(2) = 8, f(-3) = 4, f(5) = 12, and f(7) = -1, find f-1(8), f-1(4), f-1(-1) . f-1(8) = 2 f-1(4) = -3 f-1(-1) = 7

Property of Inverse Functions Let f be a one-to-one function with domain A and range B. The inverse function f-1 satisifies the following cancellation properties. for every x in A for every x in B Conversely, any function f-1 satisfying these equations is the inverse of f.

Ex 5: Show that are inverses of each other. Finding Inverses of One-to-one Functions 1. Write y = f(x). 2. Interchange the x’s and the y’s. 3. Solve the resulting equation for y. The result of this process is f-1(x).

Ex 6: Find the inverse of . Ex 7: Find the inverse of .

Ex 8:a. Sketch the graph of b. Use the graph to sketch f-1(x). c. Find the equation of f-1(x). The graph of f-1 is obtained by reflecting the graph of f over the line y = x.

Assignment S 3.7: pg 286 #5-6,8-10,18,22,28,35-37,44,52,63,64