MATH 107 UMBC Common Factors in the Numerator and Denominator Questions

MATH 107 Quiz 5MATH 107_SU21 QUIZ 5
Name:______________________
Date:_____________
Please show your work for all problems, and express your answers using exact values. Express fractions
in simplest form, including rationalizing all denominators.
Problem/Solution
1
score
Given the functions 𝑓(𝑥) = −
3
√𝑥
and 𝑔(𝑥) = 3𝑥 2 − 1, find the following values if
they exist.
a) (𝑔 ∘ 𝑓)(3)
b) (𝑓 ∘ 𝑔)(−1)
4
c) (𝑓 ∘ 𝑓) (9)
√2
d) (𝑔 ∘ 𝑔) ( 3 )
2
2𝑥
𝑥
Given the functions 𝑓(𝑥) =
and 𝑔(𝑥) =
, find and simplify the following
𝑥+2
2𝑥−1
functions, then state the domain of each in interval notation.
a) (𝑔 ∘ 𝑓)(𝑥)
b) (𝑓 ∘ 𝑔)(𝑥)
3
Determine whether the function ℎ(𝑥) =
2𝑥+9
is
𝑥−3
one-to-one
a) analytically, by testing ℎ(𝑐) = ℎ(𝑑)
b) graphically, by using the horizontal line test
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MATH 107 Quiz 5
4
3−𝑥
Given the one-to-one function 𝑓(𝑥) = 𝑥+3
a) State the domain and range of the function in interval notation.
b) Find the inverse, 𝑓 −1 (𝑥).
c) Check your answer using function composition.
5
Continuation of question 4
d) Find the vertical and horizontal asymptotes for 𝑓(𝑥) and 𝑓 −1 (𝑥).
e) Graph 𝑓(𝑥) and 𝑓 −1 (𝑥) on two separate graphs. Include the asymptotes as
dashed lines.
f)
6
Identify and label the coordinates of 3 points on the graph of 𝑓(𝑥). Reverse
the x and y coordinates, then plot these points on the graph of 𝑓 −1 (𝑥) to
confirm this is the inverse function.
The wind chill can be calculated as
𝑊 = 35.74 + 0.6215𝑇𝑎 − 35.75𝑉 0.16 + 0.4275𝑇𝑎 𝑉 0.16
where W is the wind chill temperature in degrees Fahrenheit, Ta is the air
temperature in degrees Fahrenheit, and V is the wind speed in miles per hour. The
formula is valid when 𝑇𝑎 ≤ 50℉ and 𝑉 > 3 mph.
a) If the air temperature is 36℉ and the wind speed is 14 mph, what is the
wind chill temperature? Express your answer as a complete sentence and
round to two decimal places.
b) If the air temperature is 40℉ and the meteorologist says it will feel like 32℉,
what is the wind speed? Express your answer as a complete sentence and
round to two decimal places.
7
Continuation of question 6
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MATH 107 Quiz 5
c) Write and simplify the formula for W as a function of wind speed, 𝑊(𝑉),
when the air temperature is 𝑇𝑎 = 20℉. Show your work!
d) Find the value of V when 𝑊(𝑉) = 0. Then write a sentence interpreting
what this means.
8
Find all real solutions. Check your solutions in the original equation to see which are
true and which are extraneous.
3𝑥 − 2 − √2 − 𝑥 = 𝑥 − 1
9
A $1400 check is deposited in an account earning 1.4% interest compounded
monthly.
a) Write the equation for the amount A in the account as a function of
investment time in years, t.
b) Determine how much is in the account after 5 years, 10 years, and 20 years.
Round your answers to the nearest cent.
c) Using your equation from part a), determine how long it will take for the
investment to reach $2100. Round your answer to the nearest hundredth of
a year.
10 Simplify the following expressions.
1
)
625
a) log 5 (
b) log 3 (√27)
c) log 1 (32)
2
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