Transcript note

Welcome back to Precalculus
Homework Questions?
Review from Section 1.1
Summary of Equations of Lines
General Form : Ax  By  C
Slope-Intercept Form : y  mx  b
Point-Slope Form :
y  y1  m  x  x1 
Horizontal Line : y  a
Vertical Line : 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
y  x
3
3
Precalculus:
Functions 2014/15
Objectives:
 Determine whether relations between two
variables represent functions
 Use function notation and evaluate functions
 Find the domains of functions
 Use functions to model and solve real-life
problems
 Evaluate 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.
Digital Figures, 1–6
Copyright © Houghton Mifflin
Company. All rights reserved.
Diagrammatic Representation
(Diagram)
Not a function
Copyright © 2010 Pearson Education, Inc.
Slide 1.3 - 8
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 1
Decide whether each relation represents y as a function of x.
a)
b)
Input: x
2
2
3
4
5
Output: y
1
3
5
4
1
Not a function.
2 inputs have the same
output!
Function!.
There are no 2 inputs
have the same output.
Example:
Identifying a function
Determine if y is a function of x.
(a) x = y2
(b) y = x2 – 2
Solution
(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.
lide 1.3 - 11 Copyright © 2010 Pearson Education, Inc.
Solution (continued)
(b) y = x2 – 2
lide 1.3 - 12
Each x-value determines
exactly one y-value, so y
is a function of x.
The graph shows it
passes the vertical line
test.
Copyright © 2010 Pearson Education, Inc.
Example 3: Evaluating functions.
Let
g( x)   x  4 x  1
g(2)=
2
5
g(t)= t 2  4t  1
g(x+2)=  x2  5
You Try. Evaluate the following function for the
specified values.
Let h( x)  3x 2  2 x  4
h(0)=
4
h(2)=
12
h(x+1)=
3x  8x  1
2
Example 4. Evaluating a piecewise function.
 x  1, x  0
f  x  
 x  1, x  0
2
a) find f (2)
b) find f (1)
1
2
You try.
3x 2  x, x  2
f  x  
2 x  5, x  2
a ) find f (1)
4
b) find f (2)
1
b) find f (10)
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?
2
y
x3
Because division by zero is undefined, all values
that result in division by zero are excluded from the
domain.
Can you solve this equation?
x  4
2
Why not?
So x   4 is undefined.
Radicands of even roots must be positive expressions.
Remember this to find the domain of functions
involving even roots.
Example 5 : Find the domain of each function
g(x): {(-3,0),(-1,4),(0,2),(2,2),(4,-1)}
f  x    x2  4
all real numbers
1
h( x ) 
x 5
k ( x)  3x  2
4
V   r3
3
x  5
2
x
3
r 0
3, 1,0, 2, 4
You Try: Find the domain of each function
1
f ( x)  2
x 4
x  2
1
g ( x)  2
x 4
all real numbers
k ( x) 
4  3x
4
x
3
The Difference Quotient
The difference quotient of a function f is an expression of
the form
where h ≠ 0.
lide 1.5 - 23
Copyright © 2010 Pearson Education, Inc.
f (x  h)  f (x)
h
Calculating Difference Quotients
 Difference quotients are used in Calculus to find instantaneous
rates of change.
for f ( x)  x  4 x  7,
2
a ) f (2)
b) f ( x  3)
c)
find :
3
x  2x  4
2
f ( x  h)  f ( x )
h
2x  h  4
Student Example
 Find each of the following for
f  3
f  x  1
f ( x)  2  3x  x 2
16
x  x  4
2
f  x  h  f  x
h
3  2x  h
Homework:
 Pg. 24
 7,9, 13-23 odds, 27,33,37, 43-55 odds, 83, 85

Find the domain of the function and verify
graphically.
f x   9  x 2
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
f ( x)   .0032 x 2  x  3 where y and x are measured in
feet. Will the baseball clear a 10 foot fence located 300
feet from home plate?
yes, when x=300 feet, the height of the ball is 15 feet.
Homework:
 Pg. 24
 7,9, 13-23 odds, 27,33,37, 43-55 odds, 83, 85