Name:
Practice Test #3
Test #3 is Thursday 11/14/2024
Read all instructions carefully!
You must show all your work and/or sufficient mathematical justification to receive full credit.
Good Luck!
1. Let R(x) =
9x2 − 9x
4x − 2x2
(a) What is the domain of R?
(b) Find the y-intercept (if any) of R.
(c) Write R in lowest terms.
(d) Find the x-intercepts (if any) of R.
(e) Find the vertical asymptotes of R if any.
2. Let R(x) =
2x − 4
4x + 1
(a) What is the domain of R?
(b) Find the y-intercept (if any) of R.
(c) Write R in lowest terms.
(d) Find the x-intercepts (if any) of R.
(e) Find the vertical asymptotes of R of any.
3. Find the horizontal asymptotes of the following rational functions.
1
.
x2 − 4
5x4 − x3 + 3x2 + 4
(b) h(x) =
3x4 + 22x2 − 12
(a) f (x) =
4. Consider the one-to-one rational function f (x) =
function
2x − 1
. Use algebraic methods to find the inverse
x+3
5. (a) Use algebraic methods to find the inverse function f −1 (x) of the one-to-one function f (x) =
3x − 1
.
2x + 7
(b) What is the domain of f −1 (x)?
(c) What is the range of f −1 (x)?
6. Solve the equation log3 (x + 4) = 2.
3 2
x (x + 2)
7. Write the expression log5
as a sum and/or difference of logarithms. Express powers as
(x − 3)3
factors.
(x + 1)3 (x + 2)2
8. Write the expression log
as a sum and/or difference of logarithms. Express pow(x2 + 1)3 (x − 3)3
ers as factors.
1
ln(2a + 3) as a single logarithm.
2
1
10. Write the expression 3 ln(x + 1) − 1 ln(x − 2) + ln(x − 4) as a single logarithm.
2
9. Write the expression 3 ln(5a3 ) + 3 ln(3a + 1) −
11. Graph the function f (x) = 1 + 3(x−2) . Make sure you graph at least two (exact) points points and
label any vertical or horizontal asymptotes. Clearly state the domain and range of f .
12. Graph the function f (x) = 1 + log2 (x + 1). Make sure you graph at least two (exact) points points and
label any vertical or horizontal asymptotes. Clearly state the domain and range of f .