SOLUTION: The function g(x) = 4 - 3sinx is defined for the domain π/2 ≤ x ≤ A (i)State the largest value of A for which g has an inverse (ii) For this value of A, find the value of g^

Question 1202361: The function g(x) = 4 - 3sinx is defined for the domain π/2 ≤ x ≤ A
(i)State the largest value of A for which g has an inverse
(ii) For this value of A, find the value of g^-1(3)
Answer by math_tutor2020(3835) (Show Source): You can put this solution on YOUR website!

Part (i)

Make sure your calculator is in radian mode.

The period of sin(x) is 2pi units. The curve repeats itself every 2pi units.
Half of this is 2pi/2 = pi, and this amount is added onto the left endpoint pi/2.
This is so we can determine the largest value of A to have g(x) be invertible.

A = pi/2 + pi = pi/2 + 2pi/2 = (pi+2pi)/2 = 3pi/2

Therefore, g(x) is invertible on the interval pi/2 ≤ x ≤ 3pi/2

Check out this Desmos graph
https://www.desmos.com/calculator/7uxfzjmjsk
The value of A is currently set to 3pi/2 (which is 4.712389 approximately)
If you move the slider of A around, then the blue curve will grow or shrink depending on what happens with A.

If A > 3pi/2, then the blue curve will not be one-to-one. Pieces of it will fail the horizontal line test.
Therefore, it won't be invertible for A > 3pi/2.

Answer: A = 3pi/2

=====================================================================================================

Part (ii)

The task of finding g^(-1)(3) is the same as solving g(x) = 3 when pi/2 ≤ x ≤ 3pi/2.

Use your calculator to find these two approximations:
pi/2 = 1.570796
3pi/2 = 4.712389

This means pi/2 ≤ x ≤ 3pi/2 approximates to 1.570796 ≤ x ≤ 4.712389

We're looking for a value of x between roughly 1.570796 and 4.712389, that will make g(x) = 3 true.

Let's plug in g(x) = 3 and solve for x.
g(x) = 4 - 3sin(x)
3 = 4 - 3sin(x)
-3sin(x) = 3-4
-3sin(x) = -1
sin(x) = -1/(-3)
sin(x) = 1/3
x = arcsin(1/3) or x = pi - arcsin(1/3)
x = 0.339837 or x = 2.801756
Make sure your calculator is in radian mode.

The first solution x = 0.339837 is not in the interval 1.570796 ≤ x ≤ 4.712389
The second solution x = 2.801756 is in that interval, and it is the approximate final answer.

Let's introduce the horizontal line y = 3 to the graph
https://www.desmos.com/calculator/uknjchzkwd
The sine curve and the straight line intersect at roughly (2.802,3) to help confirm our answer.
It means g(2.802) = 3 approximately.

If x = 2.802, then,
g(x) = 4 - 3sin(x)
g(2.802) = 4 - 3sin(2.802)
g(2.802) = 3.00069088973563
which is fairly close to 3.

and if x = 2.801756, then,
g(x) = 4 - 3sin(x)
g(2.801756) = 4 - 3sin(2.801756)
g(2.801756) = 3.00000072369363
Both results are really close to 3.

Answer: Approximately 2.801756

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