a. use composition to prove whether or not the functions are inverse of each other
b. express the domain of the compositions using interval notation.

⇒If , fog(x)=I(Identity),

  gof(x)=I(Identity),

then then f(x) and g(x) are inverses of each other.

fog(x)=x and gof(x)=x

fog=I and gof=I

It means f(x) and g(x) are inverses of each other.

⇒Domain of f(x)=R-{4}, R=Real Number

as⇒ x-4≠0

⇒x≠4

⇒Domain of g(x)=R-{0},R=Set of Real number

As, x≠0.

⇒Domain of the Composition

fog(x)=gof(x)=x

=Set of all Real Number(R)

(a) Increasing in the interval (-623, 2) and (623, ∞)

f is decreasing in the interval (∞, -623) and (2, - 623).

(b) Local maximum is f = 2 .

2 local minima  f = -623.

Step-by-step explanation:

f(x) = x^4 - 50x^2 + 2

Finding the derivative:

f'(x) = 4x^3 - 100x

This = 0 at  a turning point:

4x(x^2 - 25) = 0

x = 0,  x + -5, 5.

Find the second derivative:

f"(x) = 12x^2 - 100

When x = 0 f(x) = 2

When x = 0, f"(x) is negative so the point (0, 2) is a maximum .

When x = -5, f"(x) is  positive . Also positive when x = 5.

So the 2 local minimums are (-5, (-5)^4 - 50(-5)^2 + 2) = (-5, -623) and

(5, -623).