Question

f(x)=1/x-5, g(x)=5x-1/x A. Use composition to prove whether or not the functions are inverses of each other. B. Express the domain of the compositions using interval notation.

Answer 1

Answer:

Not inverse of each other

Domain : [-∞,0) U (0,5) U (5,∞]

Step-by-step explanation:

Given in the question two functions

f(x)=1/x-5

g(x)=5x-1/x

To find that each of them are inverse of each other we will use composition

f(g(x))

$\frac{1}{\frac{5x-1}{x}-5 }$

take LCM

$\frac{1}{\frac{5x-1-5x}{x}}$

5x will be cancel

$\frac{1}{\frac{-1}{x}}$

1 ÷ (-1/x)

1 × (-x/1)

-x

Now,

g(f(x))

$\frac{5\frac{1}{x-5} -1}{\frac{1}{x-5}}$

$\frac{5}{x-5} -1}$ × $5-x$

$\frac{5-x+5}{x-5} * (x-5)$

10-x

As it ended up with different answers, so f(x) and g(x) are not inverse of each other

The domain are all the possible x-values of function except x ≠ 0 and x ≠ 5

We can conclude that the domain of the composition function is

Domain : [-∞,0) U (0,5) U (5,∞]