Maths B Question? Please help me!?

Question

Could someone help me to find f^-1(x)?

f(x) = (((x^2 + 2x + 1) / 4) - 8) ^ (1/3)

Step by step working out please, but mainly the right answer!!

Then, find the domains of f(x) and f^-1(x)

Another question:

Consider the functions:

f(x) = sqrt of (5x - 2)
g(x) = sqrt of (4x - 3)

and determine the domains of f[g(x)] and g[f(x)]

please help me here with these questions!

Thank you heaps!

Answer 1

To find the inverse function, write the equation as

y = (((x^2 + 2x +1) / 4) - 8)^(1/3)

Now solve for x in terms of y:

y^3 = (x^2 + 2x + 1) / 4 -8

y^3 + 8 = (x^2 + 2x + 1) / 4

4*y^3 + 32 = (x + 1)^2

√[4*y^3 + 32] = x + 1

x = √[4*y^3 + 32] - 1

now interchange x and y:

f^-1(x) = √[4*x^3 + 32] - 1

The domain is determined by the argument of √ ≥ 0

5x-2 ≥ 0 and 4x-3 ≥ 0