Math SL1 - maya sheth
chapter 4: functions

lessons on functions

functions (4.1)

Termdefinition
relationa rule expressed with x and y that pairs domain and Range
Functiona relation that has one y value per x value
Domainset of x values
RangeSet of y values
Vertical line testtest to see if a relation is a function
ZerosA value that makes a function equal to 0

set builder notation: {x | x e r, 1 < x < 5}

operations on functions (4.2)

operationformula
sum(F + g)(x) = f(x) + g(x)
Difference (F - g)(x) = f(x) - g(x)
product(F • g)(x) = f(x) • g(x)
Quotient(F/g)(x) = f(x) / g(x) when (g(x) ≠ 0

composite of functions: f(g(x))
Where x is in the domain of g and g(x) is in the domain of f

examples

If f(x) = x + 9 and g(x) = 3x +1

(F + g)(x)

(X + 9) + (3x + 1)
X + 9 + 3x + 1
4x + 10

f(g(x))
(3x + 1) + 9
3x + 10

reflecting graphs and symmetry (4.3)

symmetryRuleEven/odd
over x-axis(x, y) » (x, -y)Neither
Over y(x, y) » (-x, y)Odd
About origin(X, y) » (-x, -y)Odd
Over y = x(X, y) » (y, x)Inverse

periodic functions (4.4)

THE PERIOD IS HOW MANY UNITS IT TAKES FOR THE FUNCTION THE RETURN TO ITS STARTING POINT IN THE CYCLE.
THE AMPLITUDE IS THE MAXIMUM AND MINIMUM: A = MAX - MIN / 2

changing values within the period function will alter the shape of the graph by moving or stretching it. See “stretching and translating graphs” section for more detail.

stretching and translating graphs (4.4)

GIVEn
f(x) = a(bx - h)² + k

Changing A stretches graph vertically
Changing b stretches graph horizontally
Changing h moves graph left or right
Changing k moves graph up or down

inverse functions (4.5)

use f^-1 to show inverse

two functions f and g are inverse functions if two statements are true:
1. G(f(x)) = x for all x in the domain of f
2. F(g(x)) = x for all x in the domain of g

to show if two functions are inverse’s algebraically, solve by switching x and y in the equation.

Example:
Find Inverse of
[ 2x + 1 = y ]
2y + 1 = x
2y = x - 1
Y = (x - 1)/2

A function that has an inverse is a one-to-one function. this can be determined through a horizontal line test. If the function is not one-to-one, it doesn’t have an inverse.