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(π‘₯) = βˆ’ 3 √π‘₯ and 𝑔(π‘₯) = 3π‘₯ 2 βˆ’ 1, find the following values if they exist. a) (𝑔 ∘ 𝑓)(3) b) (𝑓 ∘ 𝑔)(βˆ’1) 4 c) (𝑓 ∘ 𝑓) (9) √2 d) (𝑔 ∘ 𝑔) ( 3 ) 2 2π‘₯ π‘₯ Given the functions 𝑓(π‘₯) = and 𝑔(π‘₯) = , find and simplify the following π‘₯+2 2π‘₯βˆ’1 functions, then state the domain of each in interval notation. a) (𝑔 ∘ 𝑓)(π‘₯) b) (𝑓 ∘ 𝑔)(π‘₯) 3 Determine whether the function β„Ž(π‘₯) = 2π‘₯+9 is π‘₯βˆ’3 one-to-one a) analytically, by testing β„Ž(𝑐) = β„Ž(𝑑) b) graphically, by using the horizontal line test Page 1 of 3 MATH 107 Quiz 5 4 3βˆ’π‘₯ Given the one-to-one function 𝑓(π‘₯) = π‘₯+3 a) State the domain and range of the function in interval notation. b) Find the inverse, 𝑓 βˆ’1 (π‘₯). c) Check your answer using function composition. 5 Continuation of question 4 d) Find the vertical and horizontal asymptotes for 𝑓(π‘₯) and 𝑓 βˆ’1 (π‘₯). e) Graph 𝑓(π‘₯) and 𝑓 βˆ’1 (π‘₯) 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(π‘₯). Reverse the x and y coordinates, then plot these points on the graph of 𝑓 βˆ’1 (π‘₯) to confirm this is the inverse function. The wind chill can be calculated as π‘Š = 35.74 + 0.6215π‘‡π‘Ž βˆ’ 35.75𝑉 0.16 + 0.4275π‘‡π‘Ž 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 π‘‡π‘Ž ≀ 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, π‘Š(𝑉), when the air temperature is π‘‡π‘Ž = 20℉. Show your work! d) Find the value of V when π‘Š(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. 3π‘₯ βˆ’ 2 βˆ’ √2 βˆ’ π‘₯ = π‘₯ βˆ’ 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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