Download Review for Exam 2 Section 2.1 Basics of Functions and Their

Review for Exam 2
Section 2.1 Basics of Functions and Their Graphs
A _______ is a set of ordered pairs. The set of all first components of the ordered pairs is
the ______ and the set of all second components of the ordered pairs is the ______.
A _______ is one of the relations, which is a correspondence from a first set, domain, to a
second set, range, such that each element in the ______ corresponds to exactly one
element in the _____.
Example 1) {(a, 1), (a, 2), (b, 3)} is a relation, but not a function. {(a, 1), (b, 2), (c, 2)} is
a function.
Not every equation defines y as a function of x. We need to solve the equation for y.
Example 2) Determine whether each equation defines y as a function of x.
(a) x  y 3  0
(b) x  y 2  1  0
f(x) is the value of the function at the number x. Finding the function value f(x) is
evaluating the function f at x.
The _______ of a function is the graph of ordered pairs whose coordinates satisfy the
equation of the function.
Not every graph defines y as a function of x. We can decide whether the graph defines y
as a function of x using the vertical line test.
<Vertical Line Test>
If any vertical line intersects a graph in more than one point, the graph does not define y
as a function of x.
Example 3) Which of the following graphs represent functions?
y
y
x
y
x
1
x
Section 2.2 More on Functions and Their Graphs
f ( x  h)  f ( x )
The Difference Quotient is
, where h  0 .
h
Example 4) Let f ( x)  2 x 2  x  5 . Find the following difference quotients.
f (1  h)  f (1)
f ( x  h)  f ( x )
,h  0
,h  0
(a)
(b)
h
h
A piecewise function is a function that is defined by two (or more) equations over a
specified domain.
 x2  2x  3
if x  1

Example 5) For f ( x)   x  1
, find f (1) and f (2) .
5  x if x  1

<Graphs of Functions>
Let f be a function defined on an open interval I and take any x1, x2 in I with x1  x2 .
If f ( x1 )  f ( x2 ) , then f is ________ on I and the graph goes up from left to right.
If f ( x1 )  f ( x2 ) , then f is ________ on I and the graph goes down from left to right.
If f ( x1 )  f ( x2 ) , then f is ________ on I and the graph is horizontal.
Example 6) Use the graph of f to find the following.
(a) the domain of f
(b) the range of f
(c) the zeroes
(d) the y-intercept
(e) the interval(s) on which f is increasing
(f) the interval(s) on which f is constant
(g) the interval(s) on which f is decreasing
(h) f(3)-f(-2)
(i) the value of x at which f has the
relative minimum
(j) the relative maximum value
(k) the value of x for which f(x)=2
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<Even and Odd Functions>
A function f is ____ if for any x in the domain of f, f ( x)  f ( x) and the graph is
symmetric with respect to the y-axis.
A function f is ____ if for any x in the domain of f, f ( x)   f ( x) and the graph is
symmetric with respect to the origin.
Example 7) Determine whether each function is even, odd, or neither.
(a) f ( x)   x 4  1
(b) g ( x)  2 x 2  x
(c) h( x)  x 3  2 x
Section 2.3 & 2.4 Linear Functions and slopes
The slope of the line passing through two points ( x1 , y1 ), ( x2 , y2 ) is ______ _____.
The point-slope equation of the line with the slope m that passes through ( x1 , y1 ) is
_________________.
The slope-intercept equation of the line with slope m and y-intercept b is
_________________.
The general form of the equation of a line is __________, where
___________________.
Example 8) Find the slope and the intercepts of the line whose equation is
2 x  5 y  15  0 . Graph the line.
<Parallel and Perpendicular Lines>
Two lines are parallel if and only if they have the same slope.
Two lines are perpendicular if and only if the product of their slopes is ____.
Every horizontal line is perpendicular to every vertical line.
Example 9) Determine the line 4 x  y  7  0 and y  2 x  1 are parallel, perpendicular
or neither.
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Example 10) (a) Find the point-slope equation of the line passing through (2, -2) and
2
parallel to the line y  x  4 .
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(b) Find the slope-intercept equation of the line passing through (-1,3) and perpendicular
2
to the line y  x  4 .
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Section 2.5 Transformations of Functions
HSRV
Example 11) If (a, b) is a point on the graph of y=f(x), find the coordinates of the
corresponding points on the graph of y  3 f ( x  1)  4 .
Example 12) Use the graph of f to obtain the graph of g(x)  3 f (x  2) 1.

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Section 2.6 Combinations of Functions and Composite Functions
The domain of a function is the largest set of real numbers whose function value is a real
number. In particular, exclude the numbers which cause the denominator to be zero or a
square root of a negative number.
Example 13) Find the domain of f ( x) 
1 x
.
2x  3
Let f and g be two functions. Then ( f  g )( x)  __________ ,
f
( f  g )( x)  __________ , ( fg )( x)  __________ and ( )( x)  __________ .
g
The domains of f  g , f  g , fg are the intersection of the domain of f and the domain of
f
g and the domain of
is the intersection of the domain of f and the domain of g except
g
for the values of x for which g(x)=0.
The composite of f with g is denoted by f  g and is defined by ( f  g )  f ( g ( x)) . The
domain of f  g is the set of all x such that x is in the domain of g and g(x) is in the
domain of f.
The composite of g with f is denoted by g  f and is defined by ( g  f )  g ( f ( x)) . The
domain of g  f is the set of all x such that x is in the domain of f and f(x) is in the domain
of g.
f
Example 14) Let f ( x)  5  2 x and g ( x)  x 2  5 . Find ( g  f )( x) ,  (x) and
g
( g  f )( x) . Determine the domain of each function.
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Example 15) Use the graph of f and g to evaluate each function.
(a) ( f  f )(1)
(b) ( f  g )( 1)
f(x)
f
(c)  (0)
g
g(x)
(d) ( f  g )(0)
(e) ( g  g )( 2)
Section 2.7 Inverse Functions
Let f and g be two functions such that f ( g ( x))  x for every x in the domain of g and
f ( g ( x))  x for every x in the domain of f. Then g is the _______of f, denoted by ____.
The domain of f equals _______________ and the range of f equals ________________.
Not every function has an inverse, so we can decide if a function has an inverse or not
using the horizontal line test.
<Horizontal Line Test>
A function f has an inverse if there is no horizontal line that intersects the graph of f at
more than one point.
A ___________ function is a function if the graph of the function passes the horizontal
line test. Only _________ function has its inverse.
Example 16) Find the inverse of the function f ( x) 
6
x7
.
2x 1
<Graphs of f and f 1 >
If the point (a,b) is on the graph of f , then (b,a) must be on the graph of f 1 . The graph
of f 1 is the reflection of the graph of f about the line y=x.
Example 17) Use the graph of f to draw the graph of f 1 .
Section 2.8 Distance and Midpoint Formulas; Circles
The distance d between ( x1, y1 ) and ( x2 , y2 ) is ______________________.
The midpoint of the line segment with endpoints ( x1, y1 ) and ( x2 , y2 ) is _____________.
Example 18) Two points (-1,2) and (2,-1) are given. Find the distance between them and
the midpoint of the line segment.
<Circles>
The standard form of the equation of a circle with center (h,k) and radius r is
______________ and the general form of the equation of a circle is ______________.
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Example 19) (a) Find the center and the radius of the circle.
(b) Find the equation of the circle.
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