Functions please answer 3 and 4 first very urgent.

Question

1. Find the domain of   

2.  Find the period of     

3.  If the domain is restricted to the open interval    , find the range of   

4.  Imagine that you start a trip with 20 gallons of gas in your tank, that your car consumes gasoline at the constant rate of 0.05 gallons per mile, and that you drive until your tank is empty without buying any more gas. 

a.  What mileage does your car get on the trip, measured in miles per gallon?

b.  Give a formula for the volume, V, of gasoline remaining in your tank as a function of the distance, s, that you have traveled on your trip.  What is the domain of this function?

c.  Suppose you drive at the constant speed of 60 miles per hour.  Give a formula for the distance you travel, s, as a function of the number of hours, t, that you have traveled.  What is its domain?

d.  Use part (b) and (c) to give a formula for the volume of gas in your tank as a function of t.  What is its domain?

Answer 1

3.) If you graph the tan(x), you will see that at the left side of the interval, -pi/2, tan(x) → -∞, and e^-∞ =0

So the lower end of the range is (0, _____). The graph of tan(x) as you approach +pi/2 is +∞, and e^+∞ = ∞

So the upper end of the range is +infinity. The range, therefore, is (0, +∞)

4.) Imagine that you start a trip with 20 gallons of gas in your tank, that your car consumes gasoline at the constant rate of 0.05 gallons per mile, and that you drive until your tank is empty without buying any more gas. 

a. What mileage does your car get on the trip, measured in miles per gallon?

20 gallons/.05 gal/mi = 400 miles travelled. 400 miles/20gallons = 20 miles /gal.

b. Give a formula for the volume, V, of gasoline remaining in your tank as a function of the distance, s, that you have traveled on your trip. 

V(s) = 20 - s/20

What is the domain of this function?

Volume is lowest value is when empty = 0 gallons. Max volume is full = 20 gallons. Since Volume is a function of distance, this would describe the range. To describe the domain, distance s, we use what we learned in part (a). Min dist = 0 and max dist = 400,

so the domain for Volume is the distance values [0, 400].

c. Suppose you drive at the constant speed of 60 miles per hour. Give a formula for the distance you travel, s, as a function of the number of hours, t, that you have traveled. What is its domain?

s(t) = 60t (60 miles/hour * hours = miles driven= distance) 400 miles/60 mi/hour = 6.66667 hours

The domain is the interval of time values [0, 6.6667] .

d. Use part (b) and (c) to give a formula for the volume of gas in your tank as a function of t. What is its domain?

V = 20 - 60t/20 V = 20 - 3t Domain: [0,6.66666667]

Now to answer #1 and #2:

1. Find the domain of   Cannot have negative under radical, so x-2>0 or x>2

AND cannot have 0 in the denominator, so x² - x ≠0

this factors into x(x-1)≠0 so x≠0 and x-1≠0 so x≠1

The domain is where all of these restrictions overlap - on a number line it looks like this:

1. ------0------1------2–––––––––– the ---'s represent x≠0 and x≠1

These all overlap at 2 and to the right, so the domain is x≥2. Or [2, +∞)

2. Find the period of     In general f(x) = A sin(Bx + C) + D

B in your function = (2pi/3) And T = 2pi/B so the period T=2pi/[ 2pi/3 ] = 3