Analyzing Functions

Analyzing Functions From a Calculus Perspective

Function • The graph of a function, f, is the set of ordered pairs (x, f (x)) such that x is in the domain of f. • From an algebraic perspective y = f (x) • The value of a function is the directed distance, y, of the graph from a point, x, on the x-axis. • Domain: set of all x-values • Range: value of the function

Domain and Range D = [-8, -4)  (-4, ∞) D = {x|-8 < x, x ≠ -4, x ∈ ℝ} R = [-10, ∞)

Domain and Range D = [-2, 6) R = [0, 4]

Domain and Range D = (-4, 2)  (2, ∞) R = [6]  (-∞, -2)

Domain and Range D = (o, ∞) R = (-∞, 50)

Domain and Range D = (-∞, ∞) R = (-∞, ∞)

Domain and Range D = (-∞, ∞) R = (2, ∞)

Intercepts • x-intercept: where the function crosses the x-axis • Also called the zeros or roots of a function • Solved for by replacing f (x) with ‘0’ • y-intercept: where the function crosses the y-axis • Solved for by replacing x with ‘0’

Intercepts x-int: 4/3 y-int: 4/3

Intercepts x-int: 4 and 6 y-int: 4

Intercepts x-int: none y-int: √6

Symmetry: graphically • Line symmetry: when a function can be ‘folded’ along a line • Point symmetry: when a function can be rotated 180° about a point.

Symmetry: algebraically • Line symmetry: Replacing y with –y OR x with –x produces an equivalent equation • Point symmetry: Replacing y with –y AND x with –x produces an equivalent equation

Examples

Examples with algebra • x – y2 = 1 • xy = 4 • y = -x2 + 6 • x2 + y2 = 25

Even and Odd Functions • Even functions: Symmetric with respect to (WRT) the y-axis • Odd functions: Symmetric WRT the origin

Homework • pp. 19-20: # 9-13, 17-21, 25-29 odd

Analyzing Functions