Question

Given the function f, find the slope of the line tangent to the graph of f⁻¹ at the specified point on the graph of f⁻¹. F(x)=(x+1)² for x≥ -1 ;(4,1).

Answer 1

To find the slope of the line tangent to the graph of f⁻¹ at the point (4,1), we need to find the derivative of f⁻¹ and evaluate it at x = 4. The derivative of f(x) = (x+1)² is 2(x+1). Setting 2(x+1) equal to the reciprocal of the slope of the tangent line, we solve for m to find that the slope is 1/10.

Step-by-step explanation:

To find the slope of the line tangent to the graph of f⁻¹ at the given point (4,1), we need to find the derivative of f⁻¹ and evaluate it at x = 4. The function f(x) = (x+1)² for x ≥ -1.

To find the derivative of f⁻¹, we first need to find the derivative of f(x). The derivative of f(x) = (x+1)² is f'(x) = 2(x+1).

Next, we set f'(x) equal to the reciprocal of the slope of the tangent line, which is the slope of the line tangent to f⁻¹. So, set 2(x+1) = 1/m and solve for m:

m = 1/(2(x+1)), where x = 4.

Substituting x = 4, we have m = 1/(2(4+1)) = 1/10.

Therefore, the slope of the line tangent to the graph of f⁻¹ at the point (4,1) is 1/10.