Transcript Chapter 1 – Linear Relations and Functions

Chapter 1 – Linear Relations
and Functions
1.1 Relations and Functions
Definitions:
• A relation is a set of ordered pairs.
• The domain is the set of all abscissas, x-coordinates, of the ordered
pairs.
• The range is the set of all ordinates, y-coordinates, of the ordered
pairs.
• A function is a relation in which each element of the domain is
paired with exactly one element in the range.
Example: If x is a positive integer less that 6, state relation
representing the equation y = 5 + x by listing all ordered
pairs. Also state the domain and range.
Vertical line test:
A function is usually denoted by f. In function notation, the
symbol f(x) is interpreted as the value of the function f at x.
2
Find f(3) if f ( x)  4 x  2 x  5
Find f(4) if f ( x)  3x  8
What is the greatest integer function?
Find the domain of each of the following functions.
3x
2
f ( x)  2
f ( x)  x  2x  5
x 5
g ( x)  x  1
h( x)  x2  4
2.1 Compositions and Inverses of Functions
Operations with functions:
Sum
 f  g  x  f (x)  g(x)
Difference
Product
Quotient
 f  g  x  f (x)  g(x)
 f  g  x  f (x)  g(x)
f 
f ( x)
x

, g ( x)  0
  
g ( x)
g
2x
Given f(x) = x + 3 and g ( x ) 
find the values of each function. Also name all
x

5
values of x not in the domain.
 f  g  x
 f  g  x
 f  g  x 
f 
   x
g
Def: Given functions f and g, the composite function f g can be described by
the following equation.
f
g   x   f  g  x 
The domain of f g includes all of the elements x in the domain of g for which
g(x) is in the domain of f.
If f ( x)  x  1 and g ( x) 
1
, find
x 1
f
g  x
Compose f ( x)  x2 1 onto itself, this is called an iteration.
Two functions f and g are inverse functions if and only if
f
g  x   g f  x  x
Suppose f and f-1 are inverse functions. Then, f(x) = y if and
only if f-1(y) = x
Example: Given f(x) = 4x – 9, find f-1(x) and show that f and f-1
are inverse functions.
x6
Ex: Given f ( x) 
, find f-1(x).
3
Ex: Given f ( x)  x2 1 , find f-1(x).
1.3 Linear Functions and Inequalities
Solve the following equation:
x 3 x

1
4
2
Def: A linear equation has the form Ax + By + C = 0, where A and B
are not both zero. The graph is always a straight line.
Def: Value of x for which f(x) = 0 are called zeros of the function.
Graph the following
3x  2 y  6
f ( x)  3x  1
Graph the following
f ( x)  3
8  y  3x
Graph the following 2  x  y  5
1.4 Distance and Slope
We will now derive the distance formula.
The slope, m, of the line through (x1,y1) and (x2,y2) is given by the
following equation:
y y
m
2
1
x2  x1
Example: Find the distance between (4, -5) and (-2, 3).
Example: Find the slope through (4, 5) and (4, -3).
If the coordinates of P1 and P2 are (x1,y1) and (x2,y2), respectively,
 x1  x2 y1  y2 
,

2
2


then the midpoint of the line has coordinates: 
Example: Find the midpoint of the segment that has endpoints at
(5,8) and (2,6).
1.5 Forms of Linear Equations
The slope intercept form of a line is y = mx + b. The slope is m and
the y-intercept is b.
If the point with coordinates (x1,y1) lies on a line having slope m, the
point-slope form can be written as follows:
y  y1  m  x  x1 
The standard form of a linear equation is Ax + By + C = 0, where A,
B, and C are real numbers and A and B are not both zero.
Example: Write the equation 4x – 3y + 7 = 0 in slope-intercept form. Then
identify the slope and y-intercept.
Example: Write the slope-intercept form of the equation of the line through (3,7)
that has a slope of 2.
Example: Find the equation of the line through (-2,4) and has a slope of -1.
Example: Write the equation of the line passing through (3,7) and (4,-1).
1.6 Parallel and Perpendicular Lines
Two nonvertical lines are parallel if and only if their slopes
are equal. Any two vertical lines are always parallel.
Two nonvertical lines are perpendicular if and only if their
slopes are negative reciprocals.
Example: Write the standard form of the line that passes through (2,-3) and is
parallel to the line 4x – y +3 = 0.
Example: Write the standard form of the line that passes through (3,-5) and is
perpendicular to the line 2x – 3y + 6 = 0.