Download presentation

1
**Understanding Functions**

2
**A function is a rule or a correspondence that associates each x-value with exactly one y-value.**

The set of all the x-values is called the Domain of the function. For each element x in the domain, the corresponding element y is called the image of x. The set of all images of the elements of the domain is called the Range of the function.

3
**4 ways to describe a function**

Mapping Diagram Ordered pairs/Table of values Graph Rule (equation)

4
**Example: M is the Mother Function**

1. Function as a Mapping. Example: M is the Mother Function Joe Samantha Anna Ian Chelsea George Laura Julie Hilary Barbara Sue Humans Mothers

5
M: Mother function Domain of M {Joe, Samantha, Anna, Ian, Chelsea, George} Range of M {Laura, Julie, Hilary, Barbara} In function notation we can write: M(Anna) = Julie or M(George) = Barbara Also, if we are told M(x) = Hilary, That means that x must be = Chelsea

6
**For the function f below , evaluate f at the indicated values and find the Domain and Range of f**

10 11 12 13 14 15 16 1 2 3 4 5 6 7 Domain of f: Range of f: {1, 2, 3, 4, 5, 6, 7} {10, 12, 13, 15}

7
**2. Function as a Set of Ordered Pairs**

A function is a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components. In other words, you can’t have two different y-values for the same x-value.

8
**For each x, there is one related y-value**

j:{(1,-2), (2,2), (3,1), (4,-2)} p:{(0,0), (1,1)} What is h(1)? What is j(1)? What is p(1)? For what values is h(x) = 5?

9
**The mother function M can also be written as ordered pairs**

M = {(Joe, Laura), (Samantha, Laura), (Anna, Julie), (Ian, Julie), (Chelsea, Hillary), (George, Barbara) }

10
3. Function as a Graph Another way to depict a function, is to display the ordered pairs on a graph on the coordinate plane, with the x-values along the horizontal axis, and the y-values on the vertical axis.

11
**f = {(-3, -1), (-2, -3), (-1, 2), (0, -1), (1, 3), (2, 4), (3, 5)} is graphed below.**

Domain of f = {-3, -2, -1, 0, 1, 2, 3} Range of f = {-3, -1, 2, 3, 4, 5}

12
**4. Function Defined by a Rule**

Let f be a function, consisting of ordered pairs where the second element of the ordered pair is the square of the first element. Some of the ordered pairs in f are (1, 1) (2, 4), (3, 9), (4, 16),……. f is best defined by the rule f(x) = x²

13
**Function Notation f(x)**

Functions defined on infinite sets are denoted by algebraic rules. Examples of functions defined on all Real numbers f(x) = x² g(x) = 2x – h(x) = x³ The symbol f(x) represents the y-value in the Range corresponding to the Domain value x. The point (x, f(x)) belongs to the function f.

14
Evaluating functions Determine the function values (y-values) for the given x-values. 5 2 -7.5 -1 5 Undefined -11 3 Undefined If x is in the denominator, or in a square root, there will be restrictions on the Domain.

15
Graph of a function The graph of the function f(x) is the set of points (x, y) in the plane that satisfies the relation y = f(x). E.g.: The graph of the function f(x) = 2x – 1 is the graph of the equation y = 2x – 1, which is a line. Each point on the line is (x, f(x))

16
**Domain and Range from a Graph**

Remember: Domain is the set of all x-values. On a graph, it is represented by all the values from left to right. Range is the set of all the y-values. On a graph, it is represented by all the values from bottom to top. For Real numbers, we write the Domain and Range in interval notation. [ #, # ]

17
**Domain and Range from a Graph**

y 4 (-4, 2) 4 x -4 -4 Domain: x [-4, +[ Range: y [-3, +[

18
**The Zero of a Function y x**

The zero of a function is the place where the function hits the x-axis. It is the x-intercept. 2 -2 x y What is the zero of the function graphed at the right?

19
**The y-intercept of a Function**

The y-intercept of a function is the place where the function hits the y-axis. 2 -2 x y What is the y-intercept of the function graphed at the right?

20
**Calculating the zero and y-intercept of a function.**

Calculate the zero of a function by making the function equal to zero and solving for x. Calculate the y-intercept by finding f(0). Given f(x) = 2x + 10, find: a) the zero b) the y-intercept. f(x) = 2x + 10 = 0 f(0) = 2(0) + 10 2x = -10 = 10 y = 10 x = -5

21
**Consider the function: g(x) = x2 + 3x – 4**

Calculate the y-intercept of g: Calculate the zeros of g: g(0) = (0)2 + 3(0) – 4 = -4 g(x) = x2 + 3x – 4 = 0 (x + 4)(x – 1) = 0 x = -4 or x = 1

22
**Consider the function: g(x) = x2 + 3x – 4**

The zeros are x = -4 or x = 1 g(0) = -4

23
Sign of the function The function is positive on the interval x [-3, 5] A function is positive where the graph is above the x-axis. It’s negative where the graph is below. y positive The function is negative on the intervals x ]- , -3] [5, + [ -3 1 5 x negative

24
**Intervals of Increase or Decrease**

We need to identify where the function is increasing or decreasing 5 x y -3 1 Increasing: x ]-, 1] Decreasing: x [1, +[

25
Determine the Domain, Range, y-int, zeros, signs and intervals of increase and decrease for the following graph. 4 -4 (2, 3) (7, -2.5) x y

26
Domain: Determine the Domain, Range, y-int, zeros, signs and intervals of increase and decrease for the following graph. Range: Zeros: y-int: 4 -4 (2, 3) (7, -2.5) y Extrema (max/min): Positive: Negative: Increasing: x Decreasing:

27
**Theorem Vertical Line Test**

A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

28
y x Not a function.

29
y x Function.

30
**Is this a graph of a function?**

y 4 (2, 3) x -4

Similar presentations

OK

12.2 Functions and Graphs F(X) = x+4 where the domain is {-2,-1,0,1,2,3}

12.2 Functions and Graphs F(X) = x+4 where the domain is {-2,-1,0,1,2,3}

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google