Question

Please help!!

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.

Answer 1

The domain of the f(x) is (-∞;4)U(4;+∞).

The domain of the g(x) is (-∞;0)U(0;+∞).

The domain of the composition is (-∞;0)U(0;4)U(4;+∞).

To find out if the functions are inverse You should replace x in g(x) with(1/(x-4))

Do not have enough time to do the calculations. But for x = 1, functions seem to be inverse.

Answer 2

⇒If , fog(x)=I(Identity),

gof(x)=I(Identity),

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

`f(x)=(1)/(x-4)\\\\g(x)=(4x+1)/(x)\\\\fog(x)=f[g(x)]=f[(4x+1)/(x)]\\\\=(1)/((4x+1)/(x)-4)\\\\=(x)/(4x+1-4x)\\\\=x\\\\gof(x)=g[f(x)]\\\\g[(1)/(x-4)]\\\\g[f(x)]=((4 *1)/(x-4)+1)/((1)/(x-4))\\\\g[f(x)]=(4+x-4)/(1)\\\\g[f(x)]=x`

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)