Let f(x) = x 3 + x + 2, and let g(x) be the inverse function for f(x).

Question

Determine g(2)

 

Determine g'(2)

 

Determine the equation of the line tangent to the graph of y = g(x) at x = 2.

Answer 1

 First, notice that f(0) = 2, and f'(x) = 3x² + 1

 

So, since g(x) is the inverse function for f(x), g(2) = 0  (x and y get switched)

 

Thus, the point (2,0) is in the graph of y = g(x).

 

Slope of tangent to the graph of g(x) at the point (2,0) = g'(2) = 1/f'(0) = 1

 

Equation of tangent line to the graph of g(x) at (2,0):  y - 0 = (1)(x - 2)

 

                                                                                       y = x - 2

 

Answer 2

First, we write the inverse function.  Switch places with x and y.

 

x = y³ + y + 2   ---->  inverse function

 

Next, evaluate y when x=2.  This will give us g(2).

 

2 = y³ + y + 2

 

y³ + y  = 0

 

y(y² + 1) = 0

 

y = 0        or       y = ±i

 

Since we cannot have imaginary solution in a coordinate plane,         g(2) = 0

 

 

Then, we can implicitly derive this equation to find g'(2).

 

1 = 3y² y' + y'

 

1 = y' (3y² + 1)

 

1 / (3y² + 1) = y'

 

 

Plug in y=0 to find the numerical value of derivative.

 

1 / 1 = y'

 

1 = y' = g'(2)

 

 

Next, we can find the equation of the tangent line of g(x) and x=2 using point-slope form with the information we already know.

 

slope = 1

given point is (2, 0)

 

y - 0 = 1(x - 2)

 

y = x - 2