Assume f and g are inverses. It is given that f(x) = x3+5. Find the equation of the tangent line of the graph of g at the point where x = 13.

Question

Assume f and g are inverses. It is given that f(x) = x³+5. Find the equation of the tangent line of the graph of g at the point where x = 13.

Answer 1

f(x) = x³+ 5

The process of getting is the inverse is first, let's use y instead of f(x).

y = x³+ 5.

Then x becomes y and y becomes x.

x = y³+ 5.

Then solve for y as a function of x.

-y³ = -x+5

y³ = x - 5

y = (x - 5)^1/3

Therefore, the inverse of f(x), which is denoted as f^-1(x) is

f^-1(x) = g(x) = (x - 5)^1/3

Get the slope of the tangent line by solving g'(x) at x=13:

g'(x) = (1/3)(x - 5)^-2/3

g'(13) = (1/3)(13 - 5)^-2/3

g'(13) = (1/3)(8)^-2/3

g'(13) = (1/3)(1/4)

g'(13)= 1/12

From the equation, solve g(13):

g(13) = (13 - 5)^1/3 = (8)^1/3

g(13) = 2

Therefore the equation of the tangent line at (13,2) in point-slope form y - y₁ = m(x - x₁) is:

y - 2 = (1/12)(x - 13)