One-to-One Functions; Inverse Function

A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain. A function is not one-to-one if two different elements in the domain correspond to the same element in the range.

x1 y1 x1 y1 x2 y2 x2 x3 x3 y3 y3 One-to-one function NOT One-to-one Domain Range Domain Range One-to-one function NOT One-to-one function x1 y1 y2 x3 y3 Not a function Domain Range

M: Mother Function is NOT one-one Joe Samantha Anna Ian Chelsea George Laura Julie Hilary Barbara Sue Humans Mothers

S: Social Security function IS one-one Joe Samantha Anna Ian Chelsea George 123456789 223456789 333456789 433456789 533456789 633456789 Americans SSN

Is the function f below one – one? 10 11 12 13 14 15 16 1 2 3 4 5 6 7

Theorem Horizontal Line Test If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.

Use the graph to determine whether the function is one-to-one. Not one-to-one.

Use the graph to determine whether the function is one-to-one.

The inverse of a one-one function is obtained by switching the role of x and y

The inverse of the social security function Joe Samantha Anna Ian Chelsea George 123456789 223456789 333456789 433456789 533456789 633456789 SSN Americans

Let and Find

g is the inverse of f.

Let f denote a one-to-one function y = f(x) Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1. .

Domain of f Range of f

Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.

y = x (0, 2) (2, 0)

Finding the inverse of a 1-1 function Step1: Write the equation in the form Step2: Interchange x and y. Step 3: Solve for y. Step 4: Write for y.

Find the inverse of Step1: Step2: Interchange x and y Step 3: Solve for y