Question 1202361
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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
<a href="https://www.desmos.com/calculator/7uxfzjmjsk">https://www.desmos.com/calculator/7uxfzjmjsk</a>
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: <font color=red>A = 3pi/2</font>


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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
<a href="https://www.desmos.com/calculator/uknjchzkwd">https://www.desmos.com/calculator/uknjchzkwd</a>
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: <font color=red>Approximately 2.801756</font>
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