Question

Consider a function f(x) such that f(5x) =x-5/5x-1. Find f(x) and hence write down the domain of f(x).​

Answer 1

The domain of
`f(x)` is
`Dom \{f(x)\} = \mathbb{R}-\{1\}`.

Explanation:

Let
`f(5\cdot x ) = (x-5)/(5\cdot x-1)`, which is rearranged by algebraic means:

1)
`f(5\cdot x ) = (x-5)/(5\cdot x-1)` Given

2)
`f(5\cdot x) = (x\cdot 1 - 5)/(5\cdot x -1)` Modulative property

3)
`f(5\cdot x) = (5^(-1)\cdot (5\cdot x)-5)/(5\cdot x -1)` Existence of multiplicative inverse/Associative and commutative properties.

4)
`f(x) = (5^(-1)\cdot x - 5)/(x-1)` Composition of functions.

5)
`f(x) = ((5^(-1)\cdot x-5)\cdot (5\cdot 5^(-1)))/(x-1)` Existence of multiplicative inverse/Associative property.

6)
`f(x) = ((x-25)\cdot 5^(-1))/(x-1)` Commutative, associative and distributive properties/
`a\cdot (-b) =-a\cdot b`/Definition of subtraction

7)
`f(x) = (x-25)\cdot [5^(-1)\cdot (x-1)^(-1)]` Definition of division/Associative property

8)
`f(x) = (x-25)\cdot (5\cdot x -5)^(-1)`
`a^(c)\cdot b^(c) = (a\cdot b)^(c)`/Distributive property/
`a\cdot (-b) =-a\cdot b`/Definition of subtraction

9)
`f(x) = (x-25)/(5\cdot x - 5)` Definition of division/Result

The domain of a polynomial-based rational function consists in all values of the real set except values where denominator equals zero. The value of
`x` such that rational function becomes undefined is:

1)
`5\cdot x - 5 = 0` Given

2)
`5\cdot (x-1) = 0` Distributive property

3)
`[5\cdot (x-1)]\cdot 5^(-1) = 0\cdot 5^(-1)` Compatibility with multiplication

4)
`(x-1)\cdot (5\cdot 5^(-1)) = 0` Commutative and associative properties/
`a\cdot 0 = 0`

5)
`x-1 = 0` Existence of multiplicative inverse/Modulative property

6)
`x+[1+(-1)] = 1+0` Compatibility with addition/Commutative and associative properties

7)
`x = 1` Existence of additive inverse/Modulative property/Result

Hence, the domain of
`f(x)` is
`Dom \{f(x)\} = \mathbb{R}-\{1\}`.