One-to-One Functions Recall the definition of a function: A function is a relation (set of ordered pairs) such that for each x-value, there is _________________. And recall the graph of functions: If a relation is a function, its graph must pass the __________________. So, what do you think the definition of a one-to-one function is? A one-to-one (denoted by 1-1) function is a function such that for ____________________________ _______________. Or we can say, a 1-1 function is a relation such that not only for x-value there is a corresponding y-value, but also for ___________________________. What do you think the graph of a 1-1 function must pass? Since a 1-1 function is a function, so it must pass the Vertical Line Test. However, in order for a function to be 1-1, it also needs to pass the __________________. Which of the five graphs are 1-1 functions? ____________ Conclusion: If a function is 1-1, its graph must be strictly __________ or strictly __________.

Inverse Functions Two functions, f(x) and g(x) are called inverse functions f(g(x)) = x and g(f(x)) = x. How do we show a pair of functions are inverse functions? Examples: 1. f(x) = 2x + 4 2. f(x) = x2 + 3 (x  0) g(x) = ½x – 2 g(x) = Sketch each of the pair inverse functions on the same axes. Is there a line of symmetry and what is it? ___________________ What happen if we didn’t include the stated (or restricted) domain for example 2? What will its “inverse” graph be if we just flip it over the line y = x? Conclusion: In order for a function to have an inverse function, the function itself must be a _____ function. If not, though we still can plot the “inverse,” we can’t call it the inverse function.

Inverse Functions (cont’d) Q: Given a one-to-one function, f(x), how can we find its inverse function (which is commonly denoted by f –1(x), instead of g(x))? A: Follow this 4-step process: Step 1: Replace f(x) by y. Step 2: Interchange x and y (i.e., x becomes y, y becomes x). Step 3: Solve y in terms of x (this is the only step that really involves algebra). Step 4: Replace y by the notation f –1(x). Given f(x), find its inverse function f –1(x). 1. f(x) = 2x + 4 2. f(x) = x2 + 3 (x  0) If the graph of a 1-1 function is given, how do we sketch its inverse function? ________________________ Given the graph of a 1-1 function, sketch its inverse function 1. 2.