Question 1193871: a) Let f(x) = 2sinx(3x)+4 for X ∈ R. The range of f is k ≤ f(x) ≤ m. Find k and m.
b) Let g(x) = 5f(2x). Find the range of g.
c) The function g can be written in the form g(x) = 10sin(bx) + c. Find the value of b and c.
d. Find the period of g.
e. The equation g(x) = 12 has two solutions where π < x < 4π/3.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part A
The range of y = sin(x) is -1 ≤ y ≤ 1
We can say -1 ≤ sin(x) ≤ 1
The smallest sin(x) can get is -1. The largest is 1.
The same applies to sin(3x) as well
-1 ≤ sin(3x) ≤ 1
Let's double each of the three sides and then add 4 to all sides like shown below
-1 ≤ sin(3x) ≤ 1
2*(-1) ≤ 2sin(3x) ≤ 2*1
-2 ≤ 2sin(3x) ≤ 2
-2+4 ≤ 2sin(3x)+4 ≤ 2+4
2 ≤ 2sin(3x)+4 ≤ 6
The outputs of f(x) span from 2 to 6, including both endpoints
Answers:
k = 2
m = 6
============================================================
Part B
We found in part A that
2 ≤ f(x) ≤ 6
This also applies to f(2x) as well because changing the argument (aka the stuff buried in the sine function) does not alter the range.
This is because sin(anything) has the range -1 ≤ y ≤ 1
2 ≤ f(2x) ≤ 6
5*2 ≤ 5*f(2x) ≤ 5*6
10 ≤ 5*f(2x) ≤ 30
Answer:
The range is 10 ≤ g(x) ≤ 30
The smallest output for g(x) is 10, while the largest output is 30.
============================================================
Part C
f(x) = 2sin(3x) + 4
f(2x) = 2sin(3*2x) + 4 ... replace each x with 2x
f(2x) = 2sin(6x) + 4
g(x) = 5*f(2x)
g(x) = 5*(2sin(6x) + 4)
g(x) = 10sin(6x) + 5*4
g(x) = 10sin(6x) + 20
Answers:
b = 6
c = 20
============================================================
Part D
In the general template
y = A*sin(B(x-C)) + D
the value of B is tied directly to the period T like so
T = 2pi/B
B = 2pi/T
In this case, C = 0 which is the phase shift. In other words, we don't have a horizontal phase shift going on.
Comparing y = Asin(Bx) + D to y = 10sin(6x)+20 shows that
A = 10
B = 6
C = 0
D = 20
Now compute the period
T = 2pi/B
T = 2pi/6
T = pi/3
Answer: Period is pi/3 radians
============================================================
Part E
g(x) = 12
10sin(6x) + 20 = 12
10sin(6x) = 12-20
10sin(6x) = -8
sin(6x) = -8/10
sin(6x) = -4/5
6x = arcsin(-4/5) or 6x = pi - arcsin(-4/5)
6x = -0.927295 or 6x = 4.068888
x = -0.927295/6 or x = 4.068888/6
x = -0.154549 or x = 0.678148
These values are approximate.
Make sure your calculator is in radian mode.
Notice that
pi = 3.14159 approximately
4pi/3 = 4.18879 approximately
Those x values found above are NOT in the interval pi < x < 4pi/3 aka 3.14159 < x < 4.18879
What we can do though is add on pi/3 (the period found in part D) to get another root.
We keep doing this until we land in the interval mentioned.
-0.154549+pi/3 = 0.892649 not in the interval
0.892649+pi/3 = 1.939847 not in the interval
1.939847+pi/3 = 2.987045 not in the interval
2.987045+pi/3 = 4.034243 is in the interval
As for the other x value of x = 0.678148 we see that
0.678148 + 3*(pi/3) = 3.819741
which is in the interval stated.
I recommend using a tool like GeoGebra or Desmos to visually confirm the answers.
Answers:
3.819741
4.034243
Both are approximate
|
|
|
