Finding the Tangent Line to the Inverse of a Function at x = 5

Math Problem Statement

f(x)=2x^(7)+5x^(3)+4x-6 , Leibniz graphed the inverse of this function; 𝑓 −1 (𝑥) = 𝑔(𝑥). He also graphed the line that is tangent to 𝑔(𝑥) at 𝑥 = 5. What is the equation of this tangent line?

Solution

To find the equation of the tangent line to the inverse function $g(x) = f^{-1}(x)$ at $x = 5$ , we can use the following steps, leveraging the relationship between the derivatives of inverse functions.

Step-by-Step Solution

Find $y = f(5)$ : We need $f(5)$ $f (5)$ because this will help us find a point on $g(x)$ $g (x)$ at $x = 5$ $x = 5$ . $f(x) = 2x^7 + 5x^3 + 4x - 6$ $f (x) = 2 x^{7} + 5 x^{3} + 4 x - 6$ Substitute $x = 5$ $x = 5$ into $f(x)$ $f (x)$ : $f(5) = 2(5)^7 + 5(5)^3 + 4(5) - 6$ $f (5) = 2 (5)^{7} + 5 (5)^{3} + 4 (5) - 6$ Calculate each term individually:
- $2(5)^7 = 2 \times 78125 = 156250$
- $5(5)^3 = 5 \times 125 = 625$
- $4(5) = 20$ So, $f(5) = 156250 + 625 + 20 - 6 = 156889$ Therefore, $g(156889) = 5$ and $g(5) = y$ where ( g(5)

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Math Problem Analysis

Mathematical Concepts

Calculus
Inverse Functions
Derivatives
Tangent Lines

Formulas

Inverse function derivative formula: (f^{-1})'(x) = 1 / f'(f^{-1}(x))
Derivative of polynomial functions

Theorems

Derivative of Inverse Functions Theorem

Suitable Grade Level

College-level Calculus

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