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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.

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 class=

Respuesta :

Answer:

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.

⇒If , fog(x)=I(Identity),

  gof(x)=I(Identity),

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

 [tex]f(x)=\frac{1}{x-4}\\\\g(x)=\frac{4x+1}{x}\\\\fog(x)=f[g(x)]=f[\frac{4x+1}{x}]\\\\=\frac{1}{\frac{4x+1}{x}-4}\\\\=\frac{x}{4x+1-4x}\\\\=x\\\\gof(x)=g[f(x)]\\\\g[\frac{1}{x-4}]\\\\g[f(x)]=\frac{\frac{4 \times1}{x-4}+1}{\frac{1}{x-4}}\\\\g[f(x)]=\frac{4+x-4}{1}\\\\g[f(x)]=x[/tex]

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)

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