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:

The domain of [tex]f(x)[/tex] is [tex]Dom \{f(x)\} = \mathbb{R}-\{1\}[/tex].

Step-by-step explanation:

Let [tex]f(5\cdot x ) = \frac{x-5}{5\cdot x-1}[/tex], which is rearranged by algebraic means:

1) [tex]f(5\cdot x ) = \frac{x-5}{5\cdot x-1}[/tex]  Given

2) [tex]f(5\cdot x) = \frac{x\cdot 1 - 5}{5\cdot x -1}[/tex] Modulative property

3) [tex]f(5\cdot x) = \frac{5^{-1}\cdot (5\cdot x)-5}{5\cdot x -1}[/tex] Existence of multiplicative inverse/Associative and commutative properties.

4) [tex]f(x) = \frac{5^{-1}\cdot x - 5}{x-1}[/tex] Composition of functions.

5) [tex]f(x) = \frac{(5^{-1}\cdot x-5)\cdot (5\cdot 5^{-1})}{x-1}[/tex] Existence of multiplicative inverse/Associative property.

6) [tex]f(x) = \frac{(x-25)\cdot 5^{-1}}{x-1}[/tex]  Commutative, associative and distributive properties/[tex]a\cdot (-b) =-a\cdot b[/tex]/Definition of subtraction

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

8) [tex]f(x) = (x-25)\cdot (5\cdot x -5)^{-1}[/tex]  [tex]a^{c}\cdot b^{c} = (a\cdot b)^{c}[/tex]/Distributive property/[tex]a\cdot (-b) =-a\cdot b[/tex]/Definition of subtraction

9) [tex]f(x) = \frac{x-25}{5\cdot x - 5}[/tex] 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 [tex]x[/tex] such that rational function becomes undefined is:

1) [tex]5\cdot x - 5 = 0[/tex]  Given

2) [tex]5\cdot (x-1) = 0[/tex] Distributive property

3) [tex][5\cdot (x-1)]\cdot 5^{-1} = 0\cdot 5^{-1}[/tex] Compatibility with multiplication

4) [tex](x-1)\cdot (5\cdot 5^{-1}) = 0[/tex] Commutative and associative properties/[tex]a\cdot 0 = 0[/tex]

5) [tex]x-1 = 0[/tex] Existence of multiplicative inverse/Modulative property

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

7) [tex]x = 1[/tex] Existence of additive inverse/Modulative property/Result

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

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