Chapter 1.2 Functions

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.

Function Definition A function from 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

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)

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

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

F(x) = x 2 Domain = (-∞, ∞) Range = [0, ∞)

F(x) = x 3 Domain = (-∞, ∞) Range = (-∞, ∞)

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

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

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

Transformations Functions can be Moved up and down F(x) = x F(x) = x Shrink and stretched F(x) =.5x 2 F(x) = 2x 2

Transformations Turned upside down F(x) = - x 2 Moved left and right F(x) = (x 2 – 1) F(x) = (x 2 + 1)

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

Even Functions and Odd Functions A function y = f(x) is an EVEN FUNCTION of x if f(-x) = f(x) Y = x 2 y = x 4 Graph is symmetric about the y-axis ODD FUNCTION of x if f(-x) = - f(x) Y = x 3 Graph is symmetric about the origin For every x in the function’s domain

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)

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)).

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

Homework Pg 18 # Pg 19 # 5-11 odd, odd, 32, 37-41, 49-53