What is the domain and range of cos(1/x)?

Feb 28, 2018

Domain: $\left(- \infty , 0\right) \cup \left(0 , + \infty\right)$ Range: $\left[- 1 , + 1\right]$

Explanation:

$f \left(x\right) = \cos \left(\frac{1}{x}\right)$

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

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

Consider, $- 1 \le f \left(x\right) \le + 1 \forall x \in \mathbb{R} : x \ne 0$

And, ${\lim}_{x \to - \infty} f \left(x\right) = 1$

And, ${\lim}_{x \to + \infty} f \left(x\right) = 1$

Also, ${\lim}_{x \to 0} f \left(x\right)$ does not exist.

Hence, the range of $f \left(x\right)$ is $\left[- 1 , + 1\right]$

We may deduce these results from the graph of fx) below.

graph{cos(1/x) [-3.464, 3.465, -1.734, 1.728]}