Download One to One Function - Intermediate Algebra - Exam and more Exams Algebra in PDF only on Docsity!
Math 233B Name______________________
Intermediate Algebra
Fall 2012
Study Guide for Exam 5
The last chapter exam will be on Tuesday, December 4
th
. You are allowed to use one 3" by 5" index card on the
exam, as well as a scientific calculator.
For the exam you need to know how to do the following:
- Find the composite of two functions. 9.
( f g )( x ) f ( g ( x )) means first write the f -function, but replace all x ’s with big blank ( ). Inside the
big blank ( ) write the g -function and simplify.
- Determine whether a function is one-to-one. 9.
***** A function is 1-to-1 if it passes the horizontal and the vertical line test. (Draw a horizontal or vertical
line through the graph. If it intersects the line more than once, it fails the test.)
1-to-1 NOT 1-to-
***** Functions with odd leading exponents are 1-to-1. Example: (^) f ( x ) x ,
3 (^) f ( x ) x ,
5 (^) f ( x ) x ,… and
half the functions with even leading exponents are 1-to-1. Example: (^) f ( x ) x^2 , x 0 , f ( x ) | x |, x 0
- Find the inverse of a 1-to-1 function. 9.
Steps to find the inverse function:
a) Replace (^) f ( x )with (^) y.
b) Interchange the variables x and (^) y.
c) Solve the equation for (^) y.
d) Replace (^) y with ( )
1 (^) f x. This is the inverse of the function, f ( x ).
e) Verify (^) f ( x )and ( )
1 (^) f x are inverses by( f f )( x ) f ( f ( x )) x
1 1 (^) and( f f )( x ) f ( f ( x )) x
1 1
- Graph exponential functions. 9.
- Solve application problems involving exponential functions. 9.
- These problems will be like the one’s from the homework. Be prepared to calculate population
growth, decay, Compound interest formula:
nt
n
r A P ( 1 ) , and so on.
- Convert from exponential form to logarithmic form. 9.
Definition: If (^) y log ax , then
y x a
- Graph logarithmic functions. 9.
y x a , so we temporarily look at
x (^) y a , create the points (x,y) and then graph the
points (y,x). This is the graph of y x log a
- Use the properties of logarithms to expand or simplify logarithmic expressions. 9.
x log a log a log a
zero power:log 1 0 a
power rule: x n ax
n log a log * identity and power: a x
x log a
- Solve exponential and logarithmic equations. 9.
x y a a , then (^) x y.
- If (^) log b x log by , then (^) x y.
- Solve application problems involving logarithms. 9.
- Use the definition of logarithms to solve for the variable as the exponent.
- Use the properties of natural logarithms to simplify. 9.
- Natural exponential function:
x f ( x ) e
- Natural logarithms: log e x ln x If y ln x , then e x
y .
Properties
* ln e 1 * e x
x ln * x n x
n ln ln
- (^) ln xy ln x ln y * x y y
x ln ln ln * e x
ln x
- Use the change of base formula to calculate logarithms. 9.
***** Change of base formula: a
x x
b
b a log
log log
- Solve equations and application problems involving natural logarithms. 9.
- These problems will be like the one’s from the homework. Be prepared to calculate population
growth, Continuously compounded interest formula:
rt A Pe , Radioactive decay:
kt A Aoe , and so on.
Practice Problems
The answer to all the problems listed below, even and odd, are in the back of the book. For those of you who
have the Chapter Test Prep Video cd that came with the book, you can use it to see someone solving each of the
problems in the Chapter Tests. If you don't have it, it is available at the math lab.
- Write 10 10 , 000
4 in logarithmic form.
- Write x
3
(^) log 9 in exponential form, then solve for x.
- Expand and 2
3
7 5
log y
x .
- Write (^8) log 2 ( x 4 ) 8 log 2 x 3 log 2 ( x 1 )as a single expression.
- Write 4 log ( 1 ) [ 3 log 5 5 log ( 2 )]
2
1 2
1 2
1 x x.
- Evaluate 3 ( 10 )
log 6 .
- Evaluate
3 6 log 7 7.
- If a person invests $4000 at 6.5% compounded monthly, find the amount accrued at the end of 15 years.
- Say the expected population of a town which presently has 5000 residents can be approximated by the
formula
x y
- 1 5000 ( 1. 2 ). Find the expected population of the town in 20 years.
- Solve for x.
x 36 216
- Solve for x. log ( 12 ) 5
5 4 x
- Solve for x.
3
log 3 ( x 4 ) log 36 log 3
- Evaluate using the change of base formula. log 45. 3 9
- Evaluate using the change of base formula. log 0. 479 3
- Solve for t.
t e
0023 100 500
- Solve for y. ln y ln( x 5 ) 3
- If you invested $9,160 at a rate of 9% compounded continuously, how many years will it take to obtain
- The radioactive element, carbon 14, decays at a rate of 0.01205% per year. The amount of carbon 14 in
an object after t years is given by the function
t f t e
- 0001205 (^) () 200 when there were initially 200 grams
present. Find the amount of carbon 14 remaining after 100 years.
