If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

Verified step by step guidance

1. First, understand that the problem involves finding the tangent line to the inverse function f^−1(x) at a specific point. We are given that f(3) = 8, which means that f^−1(8) = 3.
2. Recall that the derivative of the inverse function at a point x is given by the formula: (f^−1)'(x) = 1 / f'(f^−1(x)). This formula helps us find the slope of the tangent line to the inverse function.
3. Substitute the given values into the formula: Since f^−1(8) = 3 and f′(3) = 7, we have (f^−1)'(8) = 1 / 7. This is the slope of the tangent line at x = 8.
4. Now, use the point-slope form of the equation of a line, which is y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. Here, the point is (8, 3) and the slope m is 1/7.
5. Substitute the values into the point-slope form: y - 3 = (1/7)(x - 8). This equation represents the tangent line to y = f^−1(x) at x = 8.

Key Concepts

Inverse Function

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^−1(y) takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Derivative of Inverse Functions

The derivative of an inverse function can be found using the formula (f^−1)'(y) = 1 / f'(x), where y = f(x). This relationship is crucial for finding the slope of the tangent line to the inverse function at a given point, as it connects the rates of change of the original and inverse functions.

Tangent Line Equation

The equation of a tangent line at a point on a curve can be expressed as y - f(a) = f'(a)(x - a), where (a, f(a)) is the point of tangency. This equation uses the slope of the function at that point and the coordinates to describe the line that just touches the curve without crossing it.