If f(x)=cosx + 3
how do I find f inverse(1)?
Thanks
y = cos(x) + 3
the inverse of this is
x = cos(y) + 3
solve for y and you have your inverse
The cos function only has a range of [-1,1], so the range of f(x) is [2,4]. this means f inverse of 1 doesn't exist.
I didn't understand this.
I have to find f inverse(1) and the derivative of f inverse(1)
Answers are 0 and 1/3 respectively.
This was your question
If f(x)=cosx + 3
how do I find f inverse(1)?
Is this cos(x+3) or cos(x) + 3, there is a difference. The first one is a shift up of the cosine function. The second is a shift to the right by 3 units.
When you want to know the derivate of f inverse calculate f' , take the reciprocal and evaluate at the point.
I'm also assuming you're using radians, not degrees.
If f(x)=cos(x) + 3 then f'(x)=-sin(x)
so f'-1(x)= -1/sin(x)
I am really really sorry.
It is f(x) = cosx + 3x
I have to find f inverse(1) and derivative of f inverse(1)
Now this is a completely different function altogether, and it's defined for all x. As x goes from (-infty,+infty) f(x) goes from (-infty,+infty).
To find f inverse 1 you want
1 = cos(x) +3x
you need some kind of root algroithm to solve this, but if you graph it you'll find x=0 then f(x)=1
f'(x)=-sin(x) + 3 so f'-1(x) = 1/(-sin(x) + 3)
You can also see f'(0)=3 so
f'-1(x) = 1/3
So f inverse(x) will be 1/(-sinx+3)
The derivative of f inverse(x) will be
cosx/(-sinx+3)²
That means f inverse(1) is 1/(-sin(1)+3) ? (should get 0)
And derivative of f inverse(1) is cos(1)/(-sin(1)+3)²? (ans: 1/3)
Am I doing right?
No, the derivative of the inverse function is the reciprocal of the derivative.
f inverse for this function doesn't have an elementary inverse function because of the cosine function.
When I reviewed my post I suspected there could be problems reading the text due to the font style.
Let's use an uppercase F for the function. Then we want
F'(x) and F'<su>-1(x), the derivative of the inverse function.
You want F<su>-1(1) which I said was x=0. You also wanted F'<su>-1(0), so I said to calculate F'<su>1(x) and reciprocate it.
F'<su>1(0)=3 so you should be able to see how the answer was obtained now.
I see my tags are incorrect. This is what it should be.
First, that is not how to find the inverse of a function.
No, the derivative of the inverse function is the reciprocal of the derivative.
f inverse for this function has an elementary inverse function, but it requires results you haven't had yet.
When I reviewed my post I suspected there could be problems reading the text due to the font style.
Let's use an uppercase F for the function. Then we want
F'(x) and F'-1(x), the derivative of the inverse function.
You want F-1(1) which I said was x=0. You also wanted F'-1(0), so I said to calculate F'1(x) and reciprocate it.
F'1(0)=3 so you should be able to see how the answer was obtained now.
Once again I see an error.
Calculate F'(x) and take the reciprocal of that to find the derivative of the inverse function.
My apologies for the confusion. Let's clarify the steps for finding f inverse(1) and its derivative:
1. Start with the equation f(x) = cos(x) + 3x.
2. To find f inverse(1), set f(x) equal to 1 and solve for x:
1 = cos(x) + 3x.
This equation does not have an elementary solution and requires numerical methods to find an approximate value for x.
3. To find the derivative of the inverse function, start by finding the derivative of f(x):
f'(x) = -sin(x) + 3.
4. Then, take the reciprocal of the derivative to find the derivative of the inverse function:
f'^-1(x) = 1 / (f'(x)).
In this case, it would be f'^-1(x) = 1 / (-sin(x) + 3).
I apologize for the previous incorrect responses. Let me know if you have any further questions.
To find the inverse of the function f(x) = cos(x) + 3x and the derivative of the inverse at f inverse of 1, we need to follow these steps:
1. Find the inverse function by switching the x and y variables: x = cos(y) + 3y.
2. Solve for y: y = f inverse(x).
3. To find f inverse of 1, substitute x = 1 into the equation y = cos(x) + 3x and solve for y: 1 = cos(y) + 3y.
4. Unfortunately, there is no direct algebraic way to solve this equation for y, so we would need to use numerical methods or approximation techniques to find the value of y = f inverse of 1.
5. Once you have the value of f inverse of 1, you can find the derivative of the inverse by calculating the derivative of the original function f(x) = cos(x) + 3x, and then taking the reciprocal of that derivative evaluated at f inverse of 1.
In summary, the steps are:
1. Set up the equation x = cos(y) + 3y.
2. Solve for y to find f inverse(x).
3. Substitute x = 1 into the equation and solve for y to find f inverse of 1.
4. Calculate the derivative of f(x) = cos(x) + 3x.
5. Take the reciprocal of the derivative and evaluate it at f inverse of 1 to find the derivative of the inverse.