1. Functions MATH 109 - Precalculus S. Rook

2. Overview Section 1.4 in the textbook: – Relations & Functions – Functional notation – Identifying functions – Application: Difference Quotient 2

3. Relations & Functions

4. Relations vs. Functions Relation: any set of ordered pairs or coordinates Function: a special type of relation where every value of x corresponds to only one value of y – Each input produces only one output All functions are relations, but all relations are NOT functions: – Ex: {(0, 7), (0, -4)} is a relation, but NOT a function 4

5. Domain & Range Domain: the set of defined x-values or input values – Can sometimes be expressed as an interval Range: the set of defined y-values or output values – Can sometimes be expressed as an interval Thus far, we have seen equations in the form y = 4x – x is known as the independent variable and y is known as the dependent variable The independent variable (x in this case) can assume any value The dependent variable (y in this case) is determined by the value chosen for x 5

6. Domain & Range (Example) Ex 1: Identify the i) domain ii) range for the function: a) {(0, 4), (-2, -1), (1, 1), (7, 2)} b) 6

7. Functional Notation

8. Until now, we have seen only equations with the letter y f(x) is called function notation and describes a function f in terms of x – f(x) means the same as y y = 2x + 5 AND f(x) = 2x + 5 are the same equation/function To evaluate a function at a value, simply substitute the value into the function – Ex: For x = -3: f(-3) = 2(-3) + 5 = -1 8

9. Piecewise Functions Piecewise Function: a set of multiple equations each defined over specific intervals of x To evaluate a piecewise function, we must decide which expression the value of x corresponds to – Ex: Given f(x): x = 8 corresponds to 1 – x 4 x = 0 corresponds to 2x2x x = 5 corresponds to 1 – x 4 9

10. Evaluating a Function (Example) Ex 2: Given f(x) = 3x 2 – x + 2, evaluate: a)f(1) b)f(4) c)f(-2) d)f(c – 1) 10

11. Evaluating a Piecewise Function (Example) Ex 3: Given, evaluate: a)g(0) b)g(1) c)g(3) 11

12. Evaluating a Piecewise Function (Example) Ex 4: Given, evaluate: a)h(0) b)h(3) c)h(4) d)h(5) 12

13. Identifying Functions

14. Recall that a function associates exactly ONE value of y for each value of x To identify a function given an equation: – Solve the equation for y and determine whether each value of x yields one value of y To identify a function from a relation (a list of coordinate pairs): – Compare the x-coordinates and determine whether each value of x yields one value of y – Look for repeating x-coordinates 14

15. Identifying Functions (Continued) Vertical Line Test: If there exists a vertical line that crosses a graph more than once, the graph is NOT a function – This means that there exists AT LEAST ONE value of x that produces at least two different values of y – Otherwise the graph is a function 15

16. Finding the Domain of a Function Recall that the domain of a function is the set of x values or the interval for which the function is defined NO domain restrictions on a polynomial – By definition a polynomial is defined over all real numbers Types of functions to be aware of domain restrictions: – Square root (what is underneath must be ≥ 0) – Rational (denominator ≠ 0) 16

17. Identifying Functions (Example) Ex 5: Select the function from either i) or ii) – explain why: a)i){(0, 0), (3,1), (-5,1), (9, 9)} ii){(-4, 2), (-2, 3), (-4, -6), (5, -1)} b)i)y – x 2 = 4 ii)y 2 – x = 4 17

18. Identifying a Function (Example) Ex 6: Select the function from either i) or ii) – explain why: i)ii) 18

19. Finding the Domain of a Function (Example) Ex 7: Find the domain of the function: a)b) c)Surface area ofd) a cube: S = 6s 2 19

20. Application: Difference Quotient

21. Difference Quotient Given a function f, is called the difference quotient Allows us to study how f changes as we allow x to vary – You WILL see this again in Calculus – For this class, just be able to calculate the difference quotient and leave it in simplest form 21

22. Difference Quotient (Example) Ex 8: Find the difference quotient – simplify the answer: a) b) 22

23. Summary After studying these slides, you should be able to: – Understand the difference between a relation and a function – State the domain and range of a function given coordinate pairs or a graph – Evaluate functions and piecewise functions – Identify the domain of a function in equation form – Calculate a difference quotient Additional Practice – See the list of suggested problems for 1.4 Next lesson – Analyzing Graphs of Functions (Section 1.5) 23