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July 23, 2014

July 23, 2014

Posted by **David** on Monday, June 14, 2010 at 2:44pm.

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 -
**Reiny**, Monday, June 14, 2010 at 5:14pm1. 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 -
**Reiny**, Monday, June 14, 2010 at 5:17pm2. The period of E = 7+4.4cos(π/35)t

= 2π/(π/35)

= 70

- Math -
**Reiny**, Monday, June 14, 2010 at 5:27pm3. 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 -
**Reiny**, Monday, June 14, 2010 at 5:28pmlastly, .....

27y^4-147y^2

= 3y^2(9y^2 - 49)

= 3y^2(3y+7)(3y-7)

- Math -
**David**, Monday, June 14, 2010 at 7:48pmThanks

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