Chapter 1.2 Functions

1. Chapter 1.2 Functions

2. Function Application • The value of one variable often depends on the values of another. • The area of a circle depends on its radius • The amount of interest generated in your bank, I, account accumulates depends on the interest rate, r. • We call, I, the dependent variable because it is dependent upon r the independent variable. • In terms of Calculus • X in the independent variable, the domains • Y is the dependent variable, the range.

3. Function Definition • A functionfrom a set D(domain) to a set R(range) is a rule that assigns a unique element in R to each element in D. • Real life example: lets say you are constantly driving 60 mph on the highway. The x-axis represents time and the y-axis represents distance traveled. You are driving for 2 hours, you can only have one distance traveled. • Vertical Line Test

4. Functions Continued • Since y in dependent on x we can say “y is a function of x”. During calculus we can say y = f(x)

5. Domains and Ranges • Domain – The largest set of x-values for which the formula gives real y-values • Endpoints – The interval’s left and right boundaries • Closed interval – contain boundary points • Open interval – contain no boundary points

6. Parent Functions f(x) = x • Domain = (-∞ , ∞) • Range = (-∞ , ∞)

7. F(x) = x2 • Domain = (-∞ , ∞) • Range = [0, ∞)

8. F(x) = x3 Domain = (-∞ , ∞) Range = (-∞ , ∞)

9. F(x) = √x • Domain = [o, ∞) • Range = [0, ∞)

10. F(x) = 1/x • Domain = (-∞,0) U (0, ∞) • Range = (-∞,0) U (0, ∞)

11. F(x) = l x l • Domain = (-∞ , ∞) • Range = [0, ∞)

12. Transformations • Functions can be • Moved up and down • F(x) = x2 + 1 • F(x) = x2 - 1 • Shrink and stretched • F(x) = .5x2 • F(x) = 2x2

13. Transformations • Turned upside down • F(x) = - x2 • Moved left and right • F(x) = (x2 – 1) • F(x) = (x2 + 1)

14. Putting It All Together • Graph f(x) = -2( lxl +3) – 3 • Graph f(x) = (x+2) + 2

15. Even Functions and Odd Functions • A function y = f(x) is an • EVEN FUNCTION of x if f(-x) = f(x) • Y = x2 y = x 4 • Graph is symmetric about the y-axis • ODD FUNCTION of x if f(-x) = - f(x) • Y = x3 • Graph is symmetric about the origin For every x in the function’s domain

16. Piece-Wise Functions • Graph and solve the following function { -3 for x ≤ -3 F(x) = { x for -2 < x ≤ 3 { 4 for x > 4 a) f(2) b) f(1) c) f(27)

17. Composite Functions • Suppose some of the outputs of a function g can be used as inputs of a function f. We can then link g and f to form a new function whose inputs x are inputs of g and whose outputs are the numbers f(g(x)). “f of g of x” • (f o g)(x) = f(g(x)).

18. Composite Functions • Find a formula for f(g(x)) if g(x) = x2 and f(x) = x – 7. Then find f(g(2)) • F(x) = x – 7 • F(g(x)) = g(x) – 7 = x2 - 7 • We then find the value of f(g(2)) by substituting 2 for x • F(g(2)) = (2)2 – 7 = -3

19. Homework • Pg 18 # 7 - 12 • Pg 19 # 5-11 odd, 21-31 odd, 32, 37-41, 49-53