2. Functions Definition A function from a set S to a set T is a rule that assigns to each element of S a unique element of T. We write f : S → T. Let S = R, T = R, and let. This is mathematical shorthand for the rule “assign to each x S its square.” Determine whether f : R → R Is a function. We see that f is a function since it assigns to each element of S a unique element of T — namely its square. Example Solution

3. A nice, geometrical way to think about the condition that Each x in the domain has corresponding to it precisely one y value is this: graph of function not function If every vertical line drawn through a curve intersects that curve just once, then the curve is the graph of a function. Note

4. The Domain and Range x (input) (output) f The set S (called the domain of the function) The set T (called the range of the function)

5. Determine the domain of: Domain is: Examples Domain is:

6. Let. Determine the domain of f. Domain is: 1 Example Solution - - - - + + + + - - - - -

7. Let. Determine the domain of f. Domain is: 12 + + + + + - - - - - - + + + + + Example Solution

8. Note Domain is the projection of the graph on x-axis Range is the projection of the graph on y-axis

9. Let. Determine a domain and range for f which make f a function. x-axis y-axis Example

10. The graph of a function f is shown in figure. (i) Find the value of and the zeros of the function. (ii) What are the domain and range of f. zero is at Example Solution

11. Composition of Functions * Suppose that f and g are functions * The domain of g contains the range of f. * If x is in the domain of f then g may be applied to f (x). This is called g composed with f or the composition of g with f

12. Let and. Calculate g ◦ f and f ◦ g. Example Solution

13. The Inverse of a Function Let be a function. We say that f has an inverse if there is a function such that Notice that the symbol denotes a new function which we call the inverse of f.

14. Find the inverse of the function We solve the equation Example Solution

15. Find the inverse of the function We solve the equation Example Solution

16. The graph of is the reflection of the graph of f about the line (a, b) (b, a) Another useful fact is this: Since an invertible function must be one-to-one, two different x values can not correspond to the same y value. Looking at the figure, we see that this means In order for f to be invertible, no horizontal line can intersect the graph of f more than once. x y Note

17. Even function Odd function Symmetry its graph is symmetric with respect to the y-axis. its graph is symmetric about the origin.

18. Determine whether each of the following functions is even, odd neither even nor odd. even odd neither even nor odd Example