Transcript 1.3 Notes

Trig/Pre-Calculus
Opening Activity
Write the domain of the
following functions.
3
1) f ( x) 
x3
2) g ( x)  x  2
3) h( x)  2 x  3 x  5
2
Solve the following
inequalities.
4) 2 x  8  0
5) 3  x  5  9
6)  2 x  6 or 3 x  12
The graph of a function f is the collection of
ordered pairs (x, f(x)) where x is in the domain of f.
(2, –2) is on the graph of f(x) = (x – 1)2 – 3.
y
f(2) = (2 – 1)2 – 3
= 12 – 3
=–2
x
4
-4
(2, –2)
The domain of the function y = f (x) is the set of
values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of
y which correspond to the values of x in the domain.
y
Range
x
4
-4
Domain
Example: Find the domain and range of the
function f (x) = x  3 from its graph.
y
Range (–3, 0)
1
–1
Domain
The domain is [–3,∞).
The range is [0,∞).
x
A function f is:
• increasing on an interval if, for any x1 and x2 in the
interval, x1 < x2 implies f (x1) < f (x2),
• decreasing on an interval if, for any x1 and x2 in the
interval, x1 < x2 implies f (x1) > f (x2),
• constant on an interval if, for any x1 and x2 in the
interval, f (x1) = f (x2).
y
(–3, 6)
The graph of y = f (x):
• increases on (– ∞, –3),
2
• decreases on (–3, 3),
x
–2
• increases on (3, ∞).
(3, – 4)
A function value f(a) is called a relative minimum of f
if there is an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a)  f(x).
y
Relative maximum
x
Relative minimum
A function value f(a) is called a relative maximum of f
if there is an interval (x1, x2) that contains a such that
x1 < x < x2 implies f(a)  f(x).
Graphing Utility: Approximate the relative minimum of
the function f(x) = 3x2 – 2x – 1.
6
– 1.79
Zoom In:
–6
6
– 0.86
2.14
– 4.79
–6
-3.24
Zoom In:
The approximate
minimum is
(0.67, –3.33).
0.58
0.76
-3.43
Determine the relative minima and maxima of the
following function. Determine where the graph is
increasing, decreasing, and constant.
g x   x  x  6 x
3
2
A piecewise-defined function is composed of two or
more functions.
f(x) =
3 + x, x < 0
x2 + 1, x  0
Use when the value of x is less than 0.
Use when the value of x is greater or
equal to 0.
y
open circle
closed circle
(0 is not included.)
(0 is included.)
x
4
-4
A function f is even if for each x in the domain of f,
f (– x) = f (x).
f (x) =
x2
y
Symmetric with
respect to the y-axis.
f (– x) = (– x)2 = x2 = f (x)
x
f (x) = x2 is an even function.
A function f is odd if for each x in the domain of f,
f (– x) = – f (x).
f (x) = x3
f (– x) = (– x)3 = –x3 = – f (x)
y
Symmetric with
respect to the origin.
x
f (x) = x3 is an odd function.
Now we are going to graph the piecewise function
from DNA #4-6 by HAND.
 1
 2 x  6, x  4
f ( x)  

 x  5, x  4
Vertical Line Test
A relation is a function if no vertical line intersects its
graph in more than one point.
y
y
y=x–1
x = |y – 2|
x
x
4
-4
This graph does not pass
the vertical line test.
It is not a function.
4
-4
This graph passes the
vertical line test.
It is a function.
Graph this…
1)
2)

2 x  3, x  2
f ( x)  

 5  x, x  2
 3, xx  22
4,  2  x  1
f ( x)  
 x  2 , xx11
Increasing, Decreasing, and Constant Functions
A function f is increasing on an interval if
x1  x2 implies f ( x1 )  f ( x2 ).
A function f is decreasing on an interval if
x1  x2 implies f ( x1 )  f ( x2 ).
A function f is constant on an interval if
for any x1 & x2 we have f ( x1 )  f ( x2 ).
Consider…
Ex 1) f ( x)  ( x  2)  1
2
Ex 2) f ( x)  x 5
Relative Minimum and Maximum Values.
A function value f (a) is a relative minimum if
for if there exists an interval x1 , x2  containing a
such that if x1  x  x2 then f (a)  f ( x).
A function value f (a) is a relative maximum if
for if there exists an interval x1 , x2  containing a
such that if x1  x  x2 then f (a)  f ( x).
We will use a graphing utility to find the following
functions relative minima and maxima.
f ( x)  x  2 x
3
2
EVEN Functions
ODD Functions
A function is EVEN if
for every x :
A function is ODD if
for every x :
f ( x)  f ( x)
Every EVEN function
is symmetric about the
y-axis.
f ( x)   f ( x)
Every ODD function is
symmetric about the yaxis.
Now we are going to graph the piecewise function
from DNA #4-6 by HAND.
 1
 2 x  6, x  4
f ( x)  

 x  5, x  4
Graph this…
1)
2)

2 x  3, x  2
f ( x)  

 5  x, x  2
 3, xx  22
4,  2  x  1
f ( x)    2  x  1
 x  2 , xx11
5) For f x   x  2 x  9, find
2
f (4  h)  f (4)
, h  0.
h
6)
a.
b.
c.
f x   2  3x  x
f (3)
f ( x  1)
f ( x  h)  f ( x )
2
for
7) Determine if y is a function of x;
3
2 2
2 x  3x y  1  0
8) Find the domain :
3
f ( x) 
x 1
Ex 4)
The net sales for a car manufacturer were $14.61 billion in 2005
and $15.78 billion in 2006. Write a linear equation giving the net
sales y in terms of x, where x is the number of years since 2000.
Then use the equation to predict the net sales for 2007.
x  0 represents 2000
So, 5, 14.61 and 6, 15.78  are
2 pts on the line
14.61  15.78  1.17
m

 1.17
56
1
y  14.61  1.17 x  5
y  14.61  1.17 x  5.85
y  1.17 x  8.76
y  1.177   8.76
y  8.19  8.76
y  16.95
$16.95 billion dollars
for 2007
Graph the following linear functions. Graph #1 – 3 on
the same coordinate plane.
1) 5 x  y  4
2) 2 x  10 y  20
3) 10 x  2 y  6
4) y  1
5) x  3