Question

identify the graph of the given function. $f(x)=\frac{1}{x}$

Explanation:

Step1: Analyze domain

The function $f(x)=\frac{1}{x}$ is undefined when $x = 0$. So the domain is all real - numbers except $x=0$, i.e., $(-\infty,0)\cup(0,\infty)$.

Step2: Analyze behavior for $x>0$

When $x>0$, as $x$ increases, $\frac{1}{x}$ decreases. For example, when $x = 1$, $f(1)=1$; when $x = 2$, $f(2)=\frac{1}{2}$; as $x
ightarrow\infty$, $f(x)
ightarrow0^{+}$.

Step3: Analyze behavior for $x<0$

When $x<0$, as $x$ decreases (moves further to the left on the number - line), $\frac{1}{x}$ also decreases. For example, when $x=-1$, $f(-1)= - 1$; when $x=-2$, $f(-2)=-\frac{1}{2}$; as $x
ightarrow-\infty$, $f(x)
ightarrow0^{-}$.

Step4: Identify the graph type

The graph of $y = \frac{1}{x}$ is a hyperbola with two branches. One branch is in the first and third quadrants for $x>0$ and $x < 0$ respectively, and it approaches the $x$ - axis ( $y = 0$) and the $y$ - axis ($x = 0$) as asymptotes.

Answer:

The graph of $y=\frac{1}{x}$ is a hyperbola with the $x$ - axis ($y = 0$) and $y$ - axis ($x = 0$) as its asymptotes, having one branch in the first quadrant for $x>0$ and one branch in the third quadrant for $x<0$.