Given f(x) and g(x)=f^-1(x).
If f(1)=4 and f'(1)=-3, then find g'(4).
...No idea where to start with this. Please help?
Thanks much!
you don't state what type of function f(x) is, but since only 2 bits of information are given about it, let's assume it is linear, or else we would need more data
let f(x) = ax + b , where a and b are constants
f'(x) = a
but we are told f'(1) = 3
since a is a constant and f'(x) = a
a = -3
also f(1) = 4
a + b = 4
-3 + b = 4
b = 7
so f(x) = -3x + 7
then g(x) = (x-7)/-3 or -x/3 + 7/3
( I did assume you know how to take the inverse of a linear function)
g'(x) = -1/3 , independent of the value of x
thus g'(4) = -1/3
Ok, I have the idea, but there's one thing that's really bugging me that I don't get...
Why would a = -3?
To find g'(4), we need to first find the inverse function of f(x), denoted as f^(-1)(x). The inverse function reverses the roles of x and y in the original function, so if we know f(x), we can find f^(-1)(x) by switching x and y and solving for y.
Since f(1) = 4, it means that when x = 1, y = 4. Now we have a point (1, 4) on the graph of f(x). Similarly, since f'(1) = -3, it means that the slope of the tangent line to the graph of f(x) at x = 1 is -3.
To find g'(4), we need to find the slope of the tangent line to the inverse function at x = 4. Remember that the slope of the tangent line at a point is equal to the derivative of the function at that point.
Here's how we can go about finding g'(4):
Step 1: Find the equation of the tangent line to the graph of f(x) at x = 1.
We know that the slope of the tangent line is -3, and we have a point on the line (1, 4). Using the point-slope form of a line, we can write the equation of the line as:
y - 4 = -3(x - 1)
Step 2: Solve the equation for y to express it in terms of x.
Rewriting the equation, we have:
y = -3x + 7
Step 3: Switch x and y to find the equation of the tangent line to the inverse function at x = 4.
We switch x and y in the equation to find the equation of the tangent line to the inverse function at x = 4:
x = -3y + 7
Step 4: Solve the equation for y to express it in terms of x.
Now we have:
-3y = x - 7
y = -(1/3)x + 7/3
Step 5: Find the derivative of y with respect to x to get g'(4).
The derivative of y with respect to x is equal to the derivative of the inverse function, so:
g'(4) = -(1/3)
Therefore, g'(4) equals -(1/3).