MATH 107 UMBC Fractions in Simplest Form & Rationalizing Denominators Exercise

MATH 107 Quiz 2MATH 107_SU21 QUIZ 2
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
Find the distance between the two points and the midpoint of the segment
connecting them.
(√75, √2) and (−√3, −√18)
2
Given the equation 2(𝑥 − 1)2 ∓ 𝑦 = 8
a) Identify the y-intercept(s) and the x-intercept(s) if they exist.
b) Calculate the coordinates of 6 points on the graph; these can include the
intercepts calculated in part a).
3a) … continuing problem 2
b) c) Using Desmos, plot the points on the graph. Add the graph of the equation, scale
appropriately, and add labels to the x- and y-axes.
d) Determine algebraically whether the graph of this equation is symmetric about the
x-axis, y-axis, or origin. Explain your reasoning.
4
1
Evaluate the function for the following values of x: −2, − 4 , 0, 1, 2
𝑓(𝑥) =
5
√𝑥 + 2
4𝑥
State the domain of the function using interval notation.
a) 𝑓(𝑥) =
√𝑥−1
5−𝑥
Page 1 of 3
MATH 107 Quiz 2
𝑥 3 −9𝑥
b) 𝑓(𝑥) = 𝑥 2 −𝑥−6
6
The cost of a certain data plan has the following rate structure, where x is the number
of megabytes (MB) of data used on all devices in the plan each month and 𝐶(𝑥) is the
monthly cost in dollars.
0.024𝑥 + 17.50,
𝐶(𝑥) = {0.022𝑥 + 16.00,
0.020𝑥 + 15.00,
𝑥 ≤ 500
500 < 𝑥 < 1000 𝑥 ≥ 1000 a) What is the monthly cost for using 500 MB? What is the cost for 650 MB? Express your answers in complete sentences. b) How much does using 700 MB cost? Is it less expensive to use 700 MB for each of two months or to use 400 MB one month and 1000 MB the next? Explain, expressing your answers in complete sentences. 7 For the graph below, determine the coordinates of the following points. a) b) c) d) e) f) g) point A point B point C point D point B reflected about the x-axis point D reflected about the y-axis point A reflected about the origin Page 2 of 3 MATH 107 Quiz 2 8 Given the function ℎ(𝑥) = 1 , √1−𝑥 find and simplify the following: a) ℎ(0) 5 b) ℎ (9) c) ℎ(−2𝑎) d) ℎ(4𝑎 + 1) 9 Given the functions 𝑓(𝑥) = 2𝑥 − 1 and 𝑔(𝑥) = 𝑥 2 + 3, simplify the following: a) (𝑓 − 𝑔)(𝑥) b) (𝑓𝑔)(𝑥) 𝑓 c) (𝑔) (𝑥) 𝑔 𝑓 d) ( ) (𝑥) e) (𝑓 ∘ 𝑔)(𝑥), the composition of these functions 10 Given the functions 𝑓(𝑥) = 𝑥 2 − 1 and 𝑔(𝑥) = 1, identify the domain of each of the 𝑥 functions below and express your answers in interval notation. 𝑓 a) (𝑔) (𝑥) 𝑔 b) (𝑓 ) (𝑥) c) (𝑓 ∘ 𝑔)(𝑥) d) (𝑔 ∘ 𝑓)(𝑥) Page 3 of 3 MATH 107 Quiz 2 MATH 107_SU21 QUIZ 2 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 Find the distance between the two points and the midpoint of the segment connecting them. (√75, √2) and (−√3, −√18) 2 Given the equation 2(𝑥 − 1)2 + 𝑦 = 8 a) Identify the y-intercept(s) and the x-intercept(s) if they exist. b) Calculate the coordinates of 6 points on the graph; these can include the intercepts calculated in part a). 3a) … continuing problem 2 b) c) Using Desmos, plot the points on the graph. Add the graph of the equation, scale appropriately, and add labels to the x- and y-axes. d) Determine algebraically whether the graph of this equation is symmetric about the x-axis, y-axis, or origin. Explain your reasoning. 4 1 Evaluate the function for the following values of x: −2, − 4 , 0, 1, 2 𝑓(𝑥) = 5 √𝑥 + 2 4𝑥 State the domain of the function using interval notation. a) 𝑓(𝑥) = √𝑥−1 5−𝑥 Page 1 of 3 MATH 107 Quiz 2 𝑥 3 −9𝑥 b) 𝑓(𝑥) = 𝑥 2 −𝑥−6 6 The cost of a certain data plan has the following rate structure, where x is the number of megabytes (MB) of data used on all devices in the plan each month and 𝐶(𝑥) is the monthly cost in dollars. 0.024𝑥 + 17.50, 𝐶(𝑥) = {0.022𝑥 + 16.00, 0.020𝑥 + 15.00, 𝑥 ≤ 500 500 < 𝑥 < 1000 𝑥 ≥ 1000 a) What is the monthly cost for using 500 MB? What is the cost for 650 MB? Express your answers in complete sentences. b) How much does using 700 MB cost? Is it less expensive to use 700 MB for each of two months or to use 400 MB one month and 1000 MB the next? Explain, expressing your answers in complete sentences. 7 For the graph below, determine the coordinates of the following points. a) b) c) d) e) f) g) point A point B point C point D point B reflected about the x-axis point D reflected about the y-axis point A reflected about the origin Page 2 of 3 MATH 107 Quiz 2 8 Given the function ℎ(𝑥) = 1 , √1−𝑥 find and simplify the following: a) ℎ(0) 5 b) ℎ (9) c) ℎ(−2𝑎) d) ℎ(4𝑎 + 1) 9 Given the functions 𝑓(𝑥) = 2𝑥 − 1 and 𝑔(𝑥) = 𝑥 2 + 3, simplify the following: a) (𝑓 − 𝑔)(𝑥) b) (𝑓𝑔)(𝑥) 𝑓 c) (𝑔) (𝑥) 𝑔 𝑓 d) ( ) (𝑥) e) (𝑓 ∘ 𝑔)(𝑥), the composition of these functions 10 Given the functions 𝑓(𝑥) = 𝑥 2 − 1 and 𝑔(𝑥) = 1, identify the domain of each of the 𝑥 functions below and express your answers in interval notation. 𝑓 a) (𝑔) (𝑥) 𝑔 b) (𝑓 ) (𝑥) c) (𝑓 ∘ 𝑔)(𝑥) d) (𝑔 ∘ 𝑓)(𝑥) Page 3 of 3