The question(s) are as follows:
Given: f(x) = 3cos(3x+π) - 4
Find:
a) the inverse of F(x)
b) the domain and range of F(x)
c) the domain and range of F -1(x)
I got cos-1[(x-4)/9] - π = y for the inverse, although I'm not sure I did the problem right (I set x to y and y to x and then solved for y).
For the domain of F-1(x) I got [-5,5] and for the range I got [-π/3,1- π/3]
Is this right?
For cos(3x+π) = 4/3, don't I have to finish solving cos(3x+π) before I decide that there is no solution (Since 4/3 isn't between -1 and 1)? I have been doing these and other problems for so long today that I am getting more confused - not less, lol! Thanks for your help.
To find the inverse of the function f(x) = 3cos(3x+π) - 4:
a) Start by replacing f(x) with y: y = 3cos(3x+π) - 4.
b) Swap x and y: x = 3cos(3y+π) - 4.
c) Solve the equation for y: x + 4 = 3cos(3y+π).
d) Divide by 3: (x + 4)/3 = cos(3y+π).
e) Apply the inverse cosine function (cos^(-1)) to both sides: cos^(-1)((x + 4)/3) = 3y+π.
f) Solve for y: y = (cos^(-1)((x + 4)/3) - π)/3.
So, the inverse of f(x) is F^(-1)(x) = (cos^(-1)((x + 4)/3) - π)/3.
To find the domain and range of f(x) = 3cos(3x+π) - 4:
- The domain of f(x) is all real numbers because cos(3x+π) is defined for any value of x.
- The range of f(x) is [-7, 1] because the cosine function has a range of -1 to 1, so multiplying by 3 and subtracting 4 will give us a range of -7 to 1.
To find the domain and range of the inverse function F^(-1)(x):
- The domain of F^(-1)(x) is the same as the range of f(x), which is [-7, 1].
- The range of F^(-1)(x) is the same as the domain of f(x), which is all real numbers.
Regarding the equation cos(3x+π) = 4/3:
- You are correct that you need to solve cos(3x+π) in order to determine if there is a solution. Since the range of the cosine function is -1 to 1, there is no solution for cos(3x+π) = 4/3 because 4/3 is outside this range.
It's common to get confused after working on problems for a long time. Take a break, clear your mind, and come back to the problem with a fresh perspective. Keep practicing and asking questions to improve your understanding.