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Linear Functions
Definition
A function f is linear if its domain is
a set of numbers and it can be
expressed in the form
f ( x)  mx  b
where m and b are constants and x
denotes an arbitrary element of the
domain of f.
Change and Rate of Change
Definition

If x1 and x2 are distinct members of
the domain of f, the change in f
from x1 to x2 is f(x2) – f(x1). The
rate of change of f over the interval
from x1 to x2 is
f ( x2 )  f ( x1 )
x2  x1
Notation


Let Dx = x2 – x1 denote the change
in x. Let Df =f(x2) – f(x1) denote
the change in f.
The rate of change is the ratio
Df
Dx
Exercise



For real numbers x, let f ( x)  x .
Find the change in f from x1 = 1 to
x2 = 4.
Find the rate of change of f over the
interval from 0 to 3 .
Find a general formula for the rate
of change over the interval from x1
to x2 for any x1 and x2.
2
A Characterization of Linear Functions
A function from the real numbers to
the real numbers is linear if and
only if its rate of change is the same
for all intervals. If so, the rate of
change is the constant m in the
formula
f ( x)  mx  b
Graphs of Linear Functions
Straight Lines
Two distinct points ( x0 , y0 ) and ( x1 , y1 )
in the plane determine one and only one
straight line
Point-Slope Form
Let ( x0 , y0 ) and ( x1 , y1 ) be two
distinct points in the plane.
Case 1: x0  x1
Set
y1  y0
m
x1  x0
Equation:
or
(slope)
y  y0  m( x  x0 )
y  y0  m( x  x0 )
Case 2:
x0  x1  c
Equation: x = c.
Point-Slope Form
Suppose it is known that a line
passes through the point with
coordinates ( x0 , y0 ) and that it has
slope m. Then the equation of the
line is
y  y0  m( x  x0 )
Slope Intercept Form



y = f(x) = mx + b
m = rate of change of f = slope of
the line = tangent of angle between
the x-axis and the line
b = f(0) = y-intercept of the line
Geometrical Interpretation
The Symmetric Form


Slope-intercept and point-slope
forms cannot handle vertical lines in
the xy plane.
Symmetric form does not select one
variable as the independent variable
and the other as the dependent
variable. c, d, and e are constants.
cx  dy  e
Exercise
The graph of a linear function is the
line whose equation is
2x  5 y  8
What is the rate of change of f?
What are f(0) and f(-2)?
Systems of Linear Equations
General Form of a Linear System of
Two Equations in Two Unknowns
ax  by  
cx  dy  
Equations in Symmetric
Form of Two Straight Lines
Three Possibilities for Solutions



The lines are not parallel and intersect in
one and only one point. That is, there is
one and only one solution of the system.
The lines are distinct but parallel and do
not intersect. There are no solutions.
The equations represent the same
straight line. There are infinitely many
solutions, one for each point on the line.
Examples:
x  2y  4
1.
3x  4 y  2
x  2y  4
2.
 2x  4 y  1
x  2y  4
3.
3x  6 y  12
The Coefficient Matrix
a b

A  
c d
The Determinant of the Coefficient
Matrix
The number
a b
c d
 ad  bc
Relationship of the Determinant to the
Question of Solutions
The linear system has a unique
solution if and only if the
determinant is different from zero.
Cramer’s Rule


b
d
x
,
a b
c d
a 
c 
y
a b
c d
Not necessarily the best
method of solution.
Exercise

Solve
2x  3y  0
x  2y  1

Answer: x=3/7, y=2/7
Inverses of Linear Functions
Example
y  f ( x)  2 x  5
Given y, solve for x:
1
1
5
x   ( y  5)   y 
2
2
2
Example (continued)
The equation
1
5
x y
2
2
defines x as a linear function of y.
This function is called the inverse of
the original function. We write
f
1
1
5
( y)   y 
2
2
Equivalence
The two equations
and
y  f (x)
1
x  f ( y)
are equivalent. One is satisfied by a
pair (x,y) if and only if the other is.
General Expression for the Inverse
Function

If f (x) = mx + b and m≠0, then
1
1
b
f ( y )  ( y  b)  y 
m
m
m
1

Note: The slope of the inverse
function is the reciprocal of the
slope of the original function.
The Graphs of the Function and Its
Inverse