MATH 150 FINAL - REVIEW
MAJOR TOPICS:
This is a list of the basic material presented in MATH 150. It may not encompass all of the
material or types of question included on the final, but it does provide a compendium of the
basic material needed to succeed on the final.
FUNCTIONS:
Composition
Domain
Range
Computing inverse functions
Functions used in modeling
Graphing
Average rate of change
Examples:
2x  1
f ( x)  ln( x  1),
g ( x) 
, h ( x )  x 2  2 x, k ( x ) 
4x  2
Find the range and domain of the above functions:
Graph the above functions.
Graph the above functions with x replaced by: x-3, 2x, 2-x.
x  3,
l ( x)  2 x 1
Find the composition of: ( g  f ), ( g  g ), ( g  h), (h  g ), ( g  k ), (h  l ), (h  h)
(For what values of x are the above compositions defined?)
Find the inverse of the following functions: f , g , l , k and h
for
x0
Find the average rate of change for each of the above functions between x=5 and x=9.
BASIC FUNCTIONS: (Know what the graph looks like; know how to establish the domain
and range)
Log, ln
Exponential functions
Basic rational functions
Functions with a discontinuity
List the domain, range, and graph of the following functions:
log 4 ( x),
4 x 1 ,
f ( x) 
x
,
x 1
x0
 x
f ( x)   2
,
2 x  1 x  0
POLYNOMIALS:
Find the zeros
Behavior of the graph near zero
Graphing
f ( x) 
1
x 1
Number of Zeros
Behavior for values of x far from the origin
Graph the following polynomials. (Pay attention to the behavior of the polynomials near the
zeros and for values of x far from the origin).
p( x)  ( x  3)3 ( x  2)2 ( x  1) x 4 , p( x)  ( x  3)2 ( x  2)2 x , p( x)  ( x  3)3 ( x  2) ( x  1) x3
Find the zeros of:
p( x)  x3  3x  2,
p( x)  x3  4 x 2  7 x  10,
p( x)  x3  4 x 2  3x  2,
p( x)  x 4  x 2  2 x  2,
p( x)  2 x3  8 x 2  9 x  9,
p( x)  x5  x3  8x 2  8,
RATIONAL FUNCTIONS:
Domain
Vertical and horizontal asymptotes
Determination of the range
Slant Asymptotes
Graphing
Expressing in the form R(X)=N(X)/D(X) = P(X)+Q(x)/D(x)
For each of the following rational functions: state the domain, range, asymptotes and sketch
the graph of the function.
1
2x  1
x3
1
r ( x)  2
,
r ( x) 
,
r ( x) 
,
r ( x)  2
,
x 1
x 1
5x  2
x 1
x 1
x 1
3x  x 2
2x  x2
r ( x)  2
, r ( x)  2
, r ( x) 
,
r ( x) 
,
x  3x  2
x  3x  2
2x  2
x 1
Express the following rational functions in the form P(X) + Q(x)/D(x)
4 x3  2 x 2  2 x  3
r ( x) 
,
2x  1
9x2  x  5
r ( x) 
,
3x 2  7 x
3x3  12 x 2  9 x  11
r ( x) 
,
x5
QUADRATIC FUNCTIONS:
Representing in standard form
Vertex (min/max values)
Intercepts
Plot
Focus of the parabola
Represent each of the quadratics in standard form. Find the intercepts, vertex, and focus of
each and graph the parabola.
f ( x)  3x 2  6 x  2
f ( x)  3x 2  6 x  10
f ( x)   x 2  4 x  4
f ( x)  3 x 2  6 x  7
f ( x)   x 2  10 x
f ( x)  x 2  x  5
EQUATIONS:
Linear
Linear Systems
With logs
With exponentials
Nonlinear Systems
Rational
1
1
5

 ,
x 1 x  2 4
x  2y  z  3
2x  5 y  6z  7
2x  3y  2z  5
x  5  x  5,
50
4
1  e x
ln( x  1)  ln( x  2)  1
x  34 x  4  0,
3
1
1
x y 
2
3
2
1
1
2x  y 
2
2
log 3 ( x  15)  log 3 ( x  1)  2
e 2 x  3e x  2  0
x2  2 y  1
x 2  5 y  29
CONICS: ELLIPSE, HYPERBOLA, CIRCLE:
Find the equation for given geometric information
Given the equation, derive geometric information (semi-major/minor axis, asymptotes,
foci, graph)
For each of the following find the graph, identify the foci, asymptotes (where appropriate), &
vertices.
4 x 2  48 y  48
4 x 2  9 y 2  36 ,
4 x 2  9 y 2  36
Find the equations of the ellipse with: Foci = (0,5) and (0,-5) and major axis of length 12
1
Find the equations of the hyperbola with: Foci = (0,8) and (0,-8) and asymptotes y   x .
2
EXPONENTIALS AND LOGS:
Properties of
Transforming equation involving log to exponentials
Graphing
Use in Modeling: Radioactive decay (Half-life), Compound interest, population
growth, etc.
Solving equations which contain terms involving these functions
Problems:
1. Find the present value of $100,000 if interest is paid at a rate of 9% per year for 5
years.
2. The population of a specie of birds is model as
5600
n(t ) 
, where t is measured in years. Find the initial population of the
0.5  27.5e  0.004t
bird. What happens to the population as time gets large. Draw a sketch of n(t).
811 / 2
10 log 5
log 1 67 4,
Find: log 7 49,
SET UP EQUATIONS TO SOLVE A GIVEN PROBLEM (WORD PROBLEMS):
Page 647 (Text) problems: 38, 39, 40, 41
PARAMETRIC EQUATIONS:
Sketch the Curve:
x  6t  4,
y  3t,
t  0,
x  t2,
y  t 4  1,
t0
INEQUALITIES:
Solve:
3 x
 1,
3 x
5 x 2  3x  3x 2  2
( x  2)( x  1)( x  3)  0,
COMPLEX NUMBERS:
Find the real and imaginary parts of each of the following:
2i
3
2  3i
3  2i

i
(3i  1)(7  3i ) ,
,
,
3  2i 3  2i
i 1
1 i
LIMITS:
1
(3  h) 1 
x2  1
x2  x  6
3
lim
,
lim
lim
h0
x2
x 1 x  1
h
x2
ALGEBRAIC EXPRESSIONS: Radicals, exponentiation laws, adding/multiplying
rational expressions.
Simplify:
 x 23

 y 12

1
 x  2  6


 y  3 

x x
x 2  2 xy  y 2 2 x 2  xy  y 2
 2
x2  y 2
x  xy  2 y 2
4
x7
4
x3
x
3

x4 x6
x
3 y