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