University of Colorado Denver
Math 1110-009 Exam 3 Study Guide - Lana
Use the Remainder and Factor Theorems
1. Use the remainder theorem to find the remainder when ( )
is divided by
Then use the factor theorem to determine whether
is a factor of ( ). If so, then write
( ) in factored form, that is, write ( ) in the form ( ) (
)(
).
.
Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function.
2. Use the rational zeros theorem to list the potential rational zeros of ( )
Find the Real Zeros of a Polynomial Function.
3. Find the real zeros of
a.
b.
and use the real zeros to factor .
( )
( )
Solve Polynomial Equations.
4. Find the real solutions of the given equations.
a.
b.
Use the Conjugate Pairs Theorem
5. Information is given about a polynomial ( ) whose coefficients are real numbers. Find the
remaining zeros of .
a. Degree 3; zeros: 3, 4 – i
b. Degree 4; zeros: I, -16 + i
c. Degree 5; zeros: 2, I, 7i
Find a Polynomial Function with Specified Zeros
6. Form a polynomial with real coefficients having degree 4 and zeros: 1, multiplicity 2, and 3i.
Simplify your answer.
Find the Complex Zeros of a Polynomial Function
7. Find the real and complex zeros of the function.
a.
b.
( )
( )
.
Form a composite function.
8. Given ( )
a.
b.
c.
d.
√ and ( )
f
g
f
g
g  (4)
f  (2)
f  (1)
g  (0)
9. For ( )
f
, find the following expressions. Simplify your answer.
and ( )
√ , find
f
g  ( x) and  g f  ( x) . Then determine whether
g  ( x)   g f  ( x) .
Find a composite function and its domain.
10. For ( )
each.
a. f
g
b. g
f
c.
f
f
and ( )
, find the following composite functions and state the domain of
d. g g
11. For ( )
and ( )
find the following composite functions and state the domain of
each. Simplify your answers. Write the domain using interval notation.
a. f g
b. g
f
c.
f
f
d. g g
Determine whether a function is one-to-one.
12. Graph each function and determine if it is one-to-one by the horizontal line test.
a.
( )
b. g ( x) 
5x  3
2x 1
c. h( x)  12 x 2  1
Obtain the graph of the inverse function from the graph of the function.
13. Sketch the graph of the inverse of the function.
The line
is provided for reference.
Find the inverse of a function defined by an equation.
14. Consider the functions f ( x)  4 x  12 and g ( x) 
a. Find f  g ( x)  .
x
3.
4
b. Find g  f ( x)  .
c. Determine whether the functions
and
are inverses of each other.
15. The function f ( x)  6 x  3 is one-to-one.
a. Find the inverse of .
b. State the domain and range of .
c. State the domain and range of
.
16. The function f ( x)  x3  4 is one-to-one.
a. Find the inverse of .
b. State the domain and range of .
c. State the domain and range of
.
17. The function f ( x)  x 2  8 where
is one-to-one.
a. Find the inverse of .
b. State the domain and range of .
c. State the domain and range of
.
18. The function f ( x) 
8x
is one-to-one.
9x  7
a. Find the inverse of .
b. Find the domain of .
c. Find the range of using
.
Evaluate and graph exponential functions.
19. Start with the graph of
and use transformations to graph
(
)
+2.
Convert from exponential to logarithmic form and vice versa.
20. Change the exponential equation to an equivalent equation involving a logarithm.
a.
b.
21. Change the logarithmic expression to an equivalent expression involving an exponent.
a. log a 5  3
b.
y  ln17
Evaluate simple logarithmic expressions without a calculator.
22. Evaluate each expression without using a calculator.
a. log5 1
b. log 1 25
c. log 9 3
5
d. ln e3
Determine the domain of a logarithmic function.
23. Find the domain of each function. Write your answer in interval notation.
a. g ( x)  ln  x  6 


b. h( x)  4  7 log8  2 
x

7
Solve simple logarithmic and exponential equations without a calculator.
24. Solve each logarithmic equation without using a calculator.
b. log 2  9 x  6   2
a. log5 25  5 x  6
c. log x 49  2
25. Solve each exponential equation without using a calculator. Write an exact answer.
a.
b.
c.
Work with properties of logarithms.
26. Use properties of logarithms to find the exact value of the expression. Do not use a calculator.
a. log 2 214
b. 2
log 2 7
c. log16 2  log16 8
d. log3 21  log3 7
27. Suppose that log a x  3 and log a y  5 . Use properties of logarithms to find the exact value of
the expression.
 
a. log a x 2 y
 y

x
b. log a 
28. Write each expression as a sum and/or difference of logarithms. Express powers as factors.
 x  x  2 

  x  7 5 


 x2 

 x7 
b. log 
a. log 2 
c. ln
3x 1  7 x
 x  4
3
29. Write the expression as a single logarithm. Simplify your answer.
a. 4log 2 u  6log 2 v


b. log 6 x 2  25  7 log 6  x  5
Use the change of base formula to evaluate logarithms.
30. Use the change-of-base formula and a calculator to evaluate each logarithm. Do not round
until the final answer. Round to the nearest thousandth as needed.
a. log3 15
b. log 1 2 9
Solve logarithmic and exponential equations.
31. Solve the equation without a calculator. Be sure to check for extraneous solutions.
a. log x  log  x  9   1
b. log 2  x  5   3  log 2  x  3
32. Solve for without a calculator. Then use a calculator to approximate your answer to 3
decimal places.
a.
b.
c.
Hint: (c) is quadratic in form.
33. Use a graphing utility to solve the equation. Round your answer to two decimal places.
e x  7 ln x