Homework Questions? Welcome back to Precalculus. Review from Section 1.1 Summary of Equations of...

Homework Questions?

Welcome back to Precalculus

Review from Section 1.1Summary of Equations of Lines

1 1

General Form :

Slope-Intercept Form :

Point-Slope Form :

Horizontal Line :

Vertical Line :

Ax By C

y mx b

y y m x x

y a

x b

Example from Section 1.1

Find the equation of the line that passes through the points (-1,-2) and (2,6).

8 2

3 3y x

Precalculus: Functions 2015/16 Objectives:

Determine whether relations between two variables represent functions

Use function notation and evaluate functions

Find the domains of functionsUse functions to model and solve real-

life problemsEvaluate difference quotients

Definition of a Function:

A function is a relation in which each element of the domain (the set of x-values, or input) is mapped to one and only one element of the range (the set of y-values, or output).

Illustration of a Function.

Slide 1.3 - 8

Diagrammatic Diagrammatic RepresentationRepresentation

Not a function

A Function can be represented several ways:

Verbally – by a sentence that states how the input is related to the output.

Numerically – in the form of a table or a list of ordered pairs.

Graphically – a set of points graphed on the x-y coordinate plane.

Algebraically – by an equation in two variables.

Example 1Decide whether each relation represents y as a function of x.

Input: x 2 2 3 4 5

Output: y 1 3 5 4 1

a) b)

Not a function.2 inputs have the same output! Function!.

There are no 2 inputs have the same output.

Slide 1.3 - 11

Example:Example: Identifying a functionIdentifying a function

(b) y = x2 – 2

Determine if y is a function of x.

SolutionSolution

(a) x = y2

(a) If we let x = 4, then y could be either 2 or –2. So, y is not a function of x. The graph shows it fails the vertical line test.

Slide 1.3 - 12

(b) y = x2 – 2

Solution Solution (continued)(continued)

Each x-value determines exactly one y-value, so y is a function of x.

The graph shows it passes the vertical line test.

Example 3: Evaluating functions.Let

g(2)=

g(t)=

g(x+2)=

g x x x( ) 2 4 1

5

2 5x

2 4 1t t

You Try. Evaluate the following function for the specified values.Let

h(0)=

h(2)=

h(x+1)=

2( ) 3 2 4h x x x

4

12

23 8 1x x

2 1, 0

1, 0

) (2)

) ( 1)

x xf x

x x

a find f

b find f

Example 4. Evaluating a piecewise function.

1

2

23 , 2

2 5, 2

) ( 1)

) (2)

) (10)

x x xf x

x x

a find f

b find f

b find f

You try.

1

4

15

Understanding Domain

Domain refers to the set of all possible input values for which a function is defined.

Can you think of a function that might be undefined for particular values?

Can you evaluate this function at x=3?

3

2

xy

Because division by zero is undefined, all valuesthat result in division by zero are excluded from the domain.

Can you solve this equation?

42 x

Radicands of even roots must be positive expressions. Remember this to find the domain of functions involving even roots.

Why not?

So

4x is undefined.

Example 5 : Find the domain of each function

g(x): {(-3,0),(-1,4),(0,2),(2,2),(4,-1)}

2 4f x x

h xx

( ) 1

5

V r 4

33

( ) 3 2k x x

3, 1,0,2,4

5x

2

3x

0r

all real numbers

You Try: Find the domain of each function

2

1( )

4f x

x

k x x( ) 4 3

2

1( )

4g x

x

2x

all real numbers

4

3x

Slide 1.5 - 23 Copyright © 2010 Pearson Education, Inc.

The Difference QuotientThe Difference QuotientThe difference quotient of a function f is an expression of the form

where h ≠ 0.

f (x h) f (x)

h

Calculating Difference Quotients

Difference quotients are used in Calculus to find instantaneous rates of change.

2for ( ) 4 7, :f x x x find

2 4x h ( ) ( )

)f x h f x

ch

) (2)a f

) ( 3)b f x 2 2 4x x

3

Student ExampleFind each of the following for f x x x( ) 2 3 2

f x h f x

h

1f x

3f 16

2 4x x

3 2x h

Homework:

Pg. 247,9, 13-23 odds, 27,33,37, 43-55

odds, 83, 85

Find the domain of the function and verify graphically.

29 xxf

Use your calculator to answer this:

A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the function where y and x are measured in feet. Will the baseball clear a 10 foot fence located 300 feet from home plate?

f x x x( ) . 0 0 3 2 32

yes, when x=300 feet, the height of the ball is 15 feet.

Homework:

Pg. 247,9, 13-23 odds, 27,33,37, 43-55

odds, 83, 85