Findig the inverse of a trigonometric function.

Question

Find the inverse of the function and also the domain. f(x) = 12sin(14x)

Answer 1

Find the inverse of:

f(x) = 12sin(14x)

Although this equation represents a function, the inverse exists only if it’s one-to-one (it must pass the horizontal line test). This requires its domain to be restricted to the general interval for inverse sine which is [‑π/2, π/2]. The unshifted base sine function range is: [-1, 1].

For this equation the domain is determined by the sine function’s argument or:

-π/2 ≤ 14x ≤ π/2

-π/28 ≤ x ≤ π/28

Domain: [-π/28, π/28]

The range is equal to the magnitude of the sine coefficient or:

Range: [-14, 14]

Reversing the equation and exchanging x and y, we get:

Given: f(x) (or y) = 12sin(14x)

12sin(14y) = x

sin(14y) = x/12

14y = arc sin (x/12)

y = f’(x)=[arc sin (x/12)]/14

The inverse function’s domain and range are the given function’s range and domain, respectively, or:

Domain: [-14, 14]

Range: [-π/28, π/28]

Answer 2

Required to find the inverse y = f(x) = 12sin(14x)

This is not a 1-1 function. However, it is 1-1 in the restricted domain [-pi/28, pi/28] with range [-12,12].

 

To find the inverse, interchange x and y

x = 12sin(14y)

x/12 = sin (14y) ........... Arc sin (x/12) = 14y

y = [Arc sin (x/12)]/14

The range of the original function is the domain of the inverse function, and the domain of the original function is the range of the inverse function.

Domain: [-12,12] 

 

 

Answer 3

To find the inverse of a function, we simply switch places with x and y.

 

f(x) = y

 

 

x = 12sin(14y)

 

 

Solve for y.

 

x / 12 = sin(14y)

 

 

Take the inverse of sine on both sides of the equation.

 

 

(1 / sin(x / 12)) / 14 = y

 

 

To find the domain, we set   sin(x / 12) equal to zero.

 

 

sin(x / 12) = 0

 

 

x / 12 = 0           ------>       x = 0

 

x / 12 = 180       ------->      x = 2160

 

x / 12 = 360      -------->      x = 4320

 

 

These angles are restrictions in the domain   0° ≤ θ ≤ 360°.