Math

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I need help with these problems for a test.
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

  • Math -

    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!

  • Math -

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

  • Math -

    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.

  • Math -

    lastly, .....

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

  • Math -

    Thanks

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