Formulas (that will be given on the test)
Things you should know
3.3 Rational Functions
(a) For a reduced rational function, how to find the x and y intercepts, vertical
asymptotes,
horizontal/slant asymptotes and put it all together in a graph.
(b) How to graph a non -reduced rational function.
(c) How to work with models of the form at/(t + b)
4.1 Exp onential functions
(a) The basic shape of the graph y = bx and thus transformations like y = C · bx.
(b) The discrete compound interest model 
.
4.2 The Natural Exponential
(a) The basic shape of the graph y = ex and transformations of it.
(b) The continuous compound interest model A = Pert.
4.3 Inverse Functions
(a) What it means for f and g to be inverses: f(g(x)) = x and g(f(x)) = x.
Graphically they
are reflections through the line y = x. The domain of f must be the range of g
and the
range of g must be the domain of f.
(b) A function is one-to-one if every output comes from only one input.
Algebraically this
means 
implies
. Graphically this is the horizontal line
test.
(c) A function has an inverse if and only it is one-to-one. To find the inverse of
a function
f(x) given by a formula set y = f -1(x) so f(y) = x. Then solve for y . For a
function given
by a table switch the inputs and outputs. For a function given graphically reflect through
y = x.
(d) What inverse means: it's really just the reverse of f. If f(a) = b then
f -1(b) = a. In
particular f(f -1(x)) = x and f -1(f(x)) = x.
4.4 Logarithms
(a) The definition 
where bx = N. This should convince you of some
basic identities
like 
and 
.
(b) The function 
is the inverse function of bx. This gives us more
identities 
and 
.
(c) The domain, range and graph of 
.
(d) Common and natural logs: 
and
.
(e) The log laws and how to use them to condense/ expand expressions
4.5 Logarithmic and Exponential Equations
(a) Change of base formula
(b) Exponential equations.
(c) Logarithmic equations. ( Solutions may be extraneous so check your answers).
(d) Simple applications (given the equation).
Some practice questions
3.3 Rational Functions
(a) Find the x and y intercepts, all asymptotes and then graph the reduced
rational functions
f(x) = (5-2x)/(x-2), g(x) = (x-3)/(x+2)2 and h(x) = (3x2 -2x-1)/(3x-5). (This
is three separate questions).
(b) Graph the non-reduced function f(x) = (x3 - x2)/(4x - 4).
(c) Textbook Section 3.3 question 49 (page 206).
4.1 Exponential functions
(a) Describe the shape of the graph of y = C · bx. You should have four different
cases depending
on what C and b are.
(b) Find the amount of money you will have after 10 years, having deposited $200
in an account
at 8% interest compounded twice a year.
4.2 The Natural Exponential
(a) Graph the function y = ex+2 -15. What are its domain, range, x and y
intercepts and end
behavior?
(b) Find the amount of money you will have after 10 years, having deposited $200
in an account
at 8% interest compounded continuously. Compare this to the amount above where
it was
compounded twice a year.
4.3 Inverse Functions
(a) Draw a function that has an inverse. Graph that inverse. Draw a function
that does not
have an inverse.
(b) Check that f(x) = 1/(x - 5) is one-to-one both algebraically and
graphically. Find f -1.
4.4 Logarithms
(a) Expand log(xy2).
(b) Condense 
.
(c) Find the domain of f(x) = log(x + 2). Graph y = f(x).
4.5 Logarithmic and Exponential Equations
(a) Calculate 
.
(b) Solve 
.
(c) Solve (3/5)x = 25/9.
(d) Solve 2000 = 540e0.06t.
(e) Solve ln x = 2x - 3 ln(1/x).
(f) Solve (1/2)6x = 2x.
(g) Solve log(x2 - 1) - log(x + 4) = log(x).
(h) The population of an endangered insect species, in millions, is given by
where t is the number of years after the year 2000. In
what year will the insect population
first fall be low 20 milion ?
Other review
(1) Workbook: review questions 103-183 starting page 218
(2) Other pages of the workbook that look relevant.
(3) Textbook exercises/chapter test.
