# What is the domain and range of the function  f(x)=5/x?

Dec 1, 2017

The domain is $x \setminus \in \mathbb{R} , x \ne 0$.
The range is $y \setminus \in \mathbb{R} , y \ne 0$.

#### Explanation:

In general, we start with the real numbers and then exclude numbers for various reasons (can't divide by zero and taking even roots of negative numbers being the main culprits).

In this case we cannot have the denominator be zero, so we know that $x \ne 0$. There's no other issues with values of $x$, so the domain is all real numbers, but $x \ne 0$.

A better notation is $x \setminus \in \mathbb{R} , x \ne 0$.

For the range, we use the fact that this is a transformation of a well known graph. Since there are no solutions to $f \left(x\right) = 0$, $y = 0$ is not in the range of the function. That's the only value the function cannot equal, so the range is $y < 0$ and $y > 0$, which can be written as $y \setminus \in \mathbb{R} , y \ne 0$.

Dec 1, 2017

Domain : $= \left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Range : $= \left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Refer to the graph attached to examine the rational function and the curve's asymptotic behavior.

#### Explanation:

A Rational Function is a function of the form $y = \frac{P \left(x\right)}{Q \left(x\right)}$, where $P \left(x\right) \mathmr{and} Q \left(x\right)$ are Polynomials and $Q \left(x\right) \ne 0$

The Domain :

When dealing with the Domain of a Rational Function, we need to locate any points of discontinuity .

As these are the points where the function is not defined, we simply set $Q \left(x\right) = 0$ to find them.

In our problem, at $\textcolor{red}{x = 0}$, the rational function is not defined. This is the point of discontinuity. The curve will exhibit asymptotic behavior on either side of it.

Hence, our Domain : $= \left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Using interval notation :

We can also write our Domain : $= \left\{x : x \in \mathbb{R} | x = 0\right\}$

That is to say the Domain includes all Real Numbers except x = 0.

Our function will continuously approach our asymptote but never quite reach that.

The Range :

To find the Range, let us make x as the subject of our function.

We will start with $y = f \left(x\right) = \frac{5}{x}$

$\Rightarrow y = \frac{5}{x}$

Multiply both sides by x to get

$\Rightarrow x y = 5$

$\Rightarrow x = \frac{5}{y}$

Like we did for the domain , we will find out for what value(s) of y does the function is undefined.

We see that it is $y = 0$

Hence, our Range : $= \left(- \infty , 0\right) \cup \left(0 , \infty\right)$

Please refer to the graph attached for a visual representation of our rational function and it's asymptotic behavior.