1. 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

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

3. 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

4. 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.

5. 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).

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

7. 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.

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