Question

f(x) = 2x - 3, g(x) = x3 + 5, then find [fog]-1 (-9) = ?

Answer 1

To find [fog]^-1(-9), we need to first find the composition of f and g, and then find the inverse of the resulting function and evaluate it at -9.

Given: f(x) = 2x - 3 g(x) = x^3 + 5

To find the composition f(g(x)), we substitute g(x) into f:

f(g(x)) = 2(g(x)) - 3 = 2(x^3 + 5) - 3 = 2x^3 + 10 - 3 = 2x^3 + 7

Now, we need to find the inverse of f(g(x)):

Let y = f(g(x)) = 2x^3 + 7

To find the inverse, we interchange x and y and solve for y:

x = 2y^3 + 7

Solving for y:

2y^3 = x - 7

y^3 = (x - 7)/2

Taking the cube root:

y = ∛((x - 7)/2)

Now, we evaluate the inverse function at -9:

[fog]^-1(-9) = ∛((-9 - 7)/2) = ∛(-16/2) = ∛(-8) = -2

Therefore, [fog]^-1(-9) = -2.