I need help with these problems for a test.

Question

1. Find the inverse of the one-to-one function.
f(x)=5x+1/4.

2. The height of the water, H, in feet, at a boat dock t hours after 7 am is given by E=7+4.4cos pi/35 t, where t is time measured in seconds. Find the period.

3. Solve the equation on the interval [0, 2pi).
a) sin^2x - cos^2x = 0

b) sin4x=(sq3)/2

4. Factor completely, or state that the polynomial is prime.
27y^4-147y^2

Answer 1

1. Here is a neat way to find the inverse of a linear function.
perform the inverse of the operations in reverse order

original: we multiply by 5 then add 1/4
inverse: we subtract 1/4, then divide by 5
f^-1(x) = (x - 1/4)/5
= (4x-1)/20

check: pick any value of x, e.g. x = 8
f(8) = 40+1/4 = 161/4

f^-1(161/4) = (4(161/4) - 1))/20
= 160/20 = 8
checks out!

Answer 2

2. The period of E = 7+4.4cos(π/35)t
= 2π/(π/35)
= 70

Answer 3

3. sin^2x - cos^2x = 0
sin^2x - (1 - sin^2x) = 0
2sin^2x = 1
sin^2x = 1/2
sin x = ± 1/√2

x = 45°, 135°, 225° or 315°
or
x = π/4, 3π/4, 5π/4 or 7π/4

4. sin 4x = √3/2
4x = 60° or 4x = 120°
or
4x = π/3 or 4x = 2π/3

so x = π/12 or x = 2π/12 = π/6

but the period of sin4x is 90° or π/2
so by adding π/2 to each of the above answers we can get more answers until we go past 2π

π/12 + π/2 = 7π/12
7π/12 + π/2 = 13π/12
13π/12 + π/2 = 19π/12 adding one more π/2 would be beyond 2π

now do the same with π/6 to get 3 more answers from that branch

altogether there will be 8 different solutions.

Answer 4

lastly, .....

27y^4-147y^2
= 3y^2(9y^2 - 49)
= 3y^2(3y+7)(3y-7)

Answer 5

Thanks