1.9
Inverse Functions
f-1(x)
Inverse functions have symmetry
about the line y = x
Inverse of ordered pairs.
Ex. f(x) = cfw_(1,4), (2,-4), (7,3)
The inverse is:
f-1(x) = cfw_(4,1), (-4,2), (3,7)
Note that the x and ys are switched.
Definition of the Invers
1.8
Combinations of Functions:
Composite Functions
The composition of the functions f and g is
( f o g )( x) = f ( g ( x)
f composed by g of x equals f of g of x
Ex. f(x) =
x
g(x) = x - 1
( f o g )(2)
( f o g )(x )= f ( g ( x) =
Find
of 2
1
= 2 1 = 1
Ex.
1.7 Transformations of Functions
Ex. 1
Shifting Points in the Plane
Shift the triangle
three units to the
right and two units
up.
2
-2
2
-2
4
What are the three
new ordered pairs?
(2,4), (5,5), (4,- 2)
Summary of Graphs of Common Functions
f(x) = c
y=x
y=
1.10
Direct Variation
and
Inverse Variation
Direct Variation
y varies directly as x
y is directly proportional to x
y = kx
k is the constant of proportionality
Inverse Variation
y varies inversely as x
y is inversely proportional to x
k
y=
x
Ex.
Find a ma
1.6 A Library of Parent Functions
Ex. 1 Write a linear function for which f(1) = 3
and f(4) = 0
First, find the slope.
0- 3
m=
=-1
4- 1
Next, use the point-slope form of the equation
of a line.
y - y1 = m( x - x1)
y - 3 = - 1( x - 1)
Function notation.
f
1.5
Analyzing
Graphs of Functions
(2,4)
Find:
a. the domain
[-1,4)
b. the range
[-5,4]
c. f(-1) = -5
d. f(2) = 4
(4,0)
(-1,-5)
Vertical Line Test for Functions
Do the graphs represent y as a function of x?
no
yes
yes
Increasing and Decreasing Functions
-2
1.4
Functions
Function - for every x there is exactly one y.
Domain - set of x-values
Range - set of y-values
Open books to page 40, example 1.
Tell whether the equations represent y as a function
of x.
a.
x2 + y = 1
Solve for y.
y = 1 x2
For every number
1.3
Linear Equations in Two Variables
Slope of a Line
rise y2 y1
m=
=
run x2 x1
Find the slope of the lines passing through
a. (-2,0) and (3,1)
b. (-1,2) and (2,2)
c. (4,-3) and (4,5)
1
Ans. ,0,
5
Point-slope Form of the Equation of a Line
Given a point
1.2 Graphs of Equations
Ex. 1 Sketch the graph of the line
y = -3x + 5
Complete the t-graph
xy
-2
-1
0
1
2
x and y-intercepts
To find the x-intercepts, let y be 0 and solve the
equation for x.
To find the y-intercept, let x be 0 and solve the
equation for
1.1 The Cartesian Plane
Ex. 1
Shifting Points in the Plane
Shift the triangle
three units to the
right and two units
up.
2
-2
2
-2
4
What are the three
new ordered pairs?
(2,4), (5,5), (4,- 2)
The Distance Formula
d = ( x2 x1 ) + ( y2 y1 )
2
What is the d