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

Answer:

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

Step-by-step explanation:

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

1) $f(5\cdot x ) = \frac{x-5}{5\cdot x-1}$  Given

2) $f(5\cdot x) = \frac{x\cdot 1 - 5}{5\cdot x -1}$ Modulative property

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

4) $f(x) = \frac{5^{-1}\cdot x - 5}{x-1}$ Composition of functions.

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

6) $f(x) = \frac{(x-25)\cdot 5^{-1}}{x-1}$  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) $f(x) = \frac{x-25}{5\cdot x - 5}$ 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 $Dom \{f(x)\} = \mathbb{R}-\{1\}$.