# What are all possible values of f(x) = 1/x^2-2x+3?

What are all possible values of f(x) = 1/x^2-2x+3?

ReportAnswer 1

$\left(- \infty , + \infty\right)$

#### Explanation:

$f \left(x\right) = \frac{1}{x} ^ 2 - 2 x + 3$

We are asked to find "all possible values" of $f \left(x\right)$

Assuming $f \left(x\right) \in \mathbb{R}$ this means the range of $f \left(x\right) : x \in \mathbb{R}$

First let's find the domain of $f \left(x\right)$

$f \left(x\right)$ is defined $\forall x \in \mathbb{R} : x \ne 0$

$\therefore$ the domain of $f \left(x\right)$ is $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$

Now consider,

${\lim}_{x \to 0 -} f \left(x\right) = + \infty$ and. ${\lim}_{x \to 0 +} f \left(x\right) = + \infty$

Also,

${\lim}_{x \to - \infty} f \left(x\right) = + \infty$

and

${\lim}_{x \to + \infty} f \left(x\right) = - \infty$

Hence the range of $f \left(x\right) = \left(- \infty , + \infty\right)$

This maybe more clearly seen from the graph of $f \left(x\right)$ below.

graph{ 1/x^2-2x+3 [-28.88, 28.83, -14.43, 14.43]}

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