Test 3 Review
Note: This review is not intended to be comprehensive, but merely to provide some ideas
and sample problems to help you prepare for the test. In particular you should definitely
be sure to review your returned homework, your class notes, and the in-class problems
we do.
When you study for this exam, practice, practice, practice! doing problems. Re-do your
old homework problems quickly. Re-do the in-class problems. Do the review problems
I’ve posted on the webpage and make sure you can do them (and not just look at the
answer and say “I can do that”). Look over your notes for concepts to remember (pay
attention to the ideas/equations I box in class on the board!) The blue boxes in the
relevant sections in your textbook highlight the MOST important ideas; read through
them.
Section 3.3: Rational Functions (textbook pg.190-201)
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Know the definition of a rational function (f(x) = n(x) / d(x))
Remember that f(x) can be written as q(x) + r(x)/d(x)
Be able to find the vertical asymptotes (VA) of a rational function
Be able to find the horizontal asymptotes (HA) of a rational function (if any)
Be able to find the slant asymptotes of a rational function (if any)
Which kinds of rational functions have HA’s and which kinds have slant As?
Be able to read off extreme values of x for the graph of a rational function
Be able to find the zeros (x-intercepts) of a rational function (n(x) = 0)
Be able to find the y-intercepts of a rational function ( set x=0, f(0)).
Be able to find the domain of a rational function
Section 4.1: Exponential functions (textbook pg. 222-228)
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Know the definition of an exponential function with base, b
Where is the horizontal asymptote of f(x) = b^x? Any vertical asymptotes?
What is the domain of an exponential function? Range?
What are the extreme values of an exponential function?
What are the x and y intercepts of y = C * (b^x)?
How do the asymptotes, intercepts, & general graph, change when shifted
up/down/left/right?
For the graph of y = C * (b)^x, what is the y-intercept?
Be able to find the equation of an exponential function given the y-intercept and
another point on the graph (solve for C and b)
Be able to use the model for compound interest; know what all the variables mean
Section 4.2: Natural Exponential functions (textbook pg. 235-244)
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What is a natural exponential function? What is e?
Where is the horizontal asymptote of f(x) = e^x? Any vertical asymptotes?
What is the domain of a natural exponential function? Range?
What are the extreme values of a natural exponential function?
What are the x and y intercepts of y = C * (e^x)?
How do the asymptotes, intercepts, & general graph, change when shifted
up/down/left/right?
For the graph of y = C * (e)^x, what is the y-intercept?
Be able to use the model for continuous compound interest; know what all the
variables mean. How does this model differ from the compound interest equation
in Section 4.3?
Newton’s Law of Cooling: Be able to use this model and solve for a variable;
know what all the variables mean.
Section 4.3: Inverse functions (textbook pg. 249-257)
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Be able to check if the functions, f(x) and g(x) are inverse functions or not ( If f
and g are inverses, then BOTH f(g(x) = x for x in domain of g, AND g(f(x)=x for
x in domain of f)
What do the graphs of a function and its inverse look like? (Reflection about y=x
What is a one-to-one function? What does it say about the inverse of f(x)?
How can you tell if a function is one-to-one? (Horizontal line test)
How can you tell if a function has an inverse or not?
Do all functions have inverses?
Be able to find the inverse of a function, f(x). (Set f(y) = x and solve for y); be
able to do these for rational functions as well (math is trickier – check your inclass notes, or pg. 256 of textbook).
Section 4.4: Logarithmic functions (textbook pg. 261-268)
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How is f(x) = b^x related to g(x) = logb(x) ?
What is the domain of g(x) = logb(x) ? How does this change if you have a graph
of y = = logb(ax + c)?
What is logb(0)? logb(1)?
How do you find the x-intercept of a logarithmic function? (Hint: you need to set
something = 0 and solve for x). Find the y-intercept?
What base is ln(x)? log(x)?
PROPERTIES YOU WILL HAVE TO KNOW COLD (MEMORIZED WELL):
o If bp = N, then N = logb(p)
o logb(PQ) = logb(P) + logb(Q)
o logb(P/Q) = logb(P) - logb(Q)
o logb(P^r) = r logb(P)
log c (x) ln( x) log( x)
o logb(x) =
=
=
(Change of Base)
log c (b) ln( b) log( b)
o logb(b) = 1
o logb(b^x) = x
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Be ableto expand logarithmic functions using these properties
Be able to compress a bunch of separate log functions into one log function
Section 4.5: Solving logarithmic and exponential functions (textbook pg. 273-279)
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Be able to use the properties above in Section 4.4 to solve for x in exponential and
logarithmic functions. Some examples (see textbook Section 4.5 for solutions):
Solve for x:
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1) 3^x = 81
1
2) ( ) 3x  8 2
2
3) 4ex  38
4) 2(3x )  6  40
5) e 2x  7  5
6) log6(2x) = 3
7) 3 + 2log4(x) = 8
8) log(x) + log(x-3) = 1
9) log(x) + log(x-15) = 2
Here are some more practice problems:
1 x2
. Find the zeros and
5x  x 2  6
y-intercepts. What is the domain? Graph this function and label all intercepts and
asymptotes.
- Write out the equations for the asymptotes of f ( x) 
- Give me an example of a function with a slant asymptote;
A function with only a horizontal (no slants) asymptote?
- Review workbook pg. 123-124 homework problems, and Section 3.3 HW probs.
- Let f ( x)  3b x ( b>1 ) and g ( x)  log 3 ( x  5)  7 . For each of the following properties
determine which (if either) of f (x) and g (x ) have this property:
1. 0 is in the domain
2. 0 is in the range
3. The function has a zero
4. The function has a vertical asymptote
5. The function has a horizontal asymptote
6. The function is one to one
7. The function has domain all real numbers
8. The function has range all real numbers
9. The function is increasing everywhere
10. The function is decreasing everywhere