# MATH 107 Quiz 5

MATH 107 Quiz 5 MATH 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 Page 1 of 3 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 Page 2 of 3 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 Page 3 of 3
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