What's the best method to graph the following function by hand...

Question

Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, it's quite imprecise. $$f(x)={|x+1|\over x}$$

Thank you!!!

Answer 1

Here is another way.

• First, draw the graph of $\frac{x + 1}{x}$. This should be pretty simple since $\frac{x + 1}{x} = 1 + \frac{1}{x}$, and you know what $\frac{1}{x}$ looks like. You also know that $\frac{x+1}{x}$ has a zero at $x = -1$.

• Second, since $|x + 1|$ switches the sign for $x < -1$, take the graph you just drew and reflect the portion with $x < -1$ across the $x$-axis. Keep the portion with $x > -1$ the same as it was before.

Answer 2

• There is a zero at $x = -1$.

• The function does not exist at $0$ and grows infinitely large near $0$.

• The function is positive when $x > 0$ and negative when $x < 0$.

• When the function hits the $x$-axis at $x = -1$ it is a sharp corner rather than a smooth bump, since the absolute value function changes abruptly.

• As $x \to \infty$, the function approaches $1$ from above.

• As $x \to -\infty$, the function approaches $-1$ from above.

Along with a few plotted points, this should give you enough information to draw a precise graph.

Answer 3

First you have to break the mod into cases when $x>-1$ and...

Let me talk of case : $\frac{x+1}{x}=1+\frac 1 x$

$1/x $ is odd function. It tends to infinity at $0^+$ and tends to $0$ at infinity.

Make graph of $\frac 1 x $ and shift it one unit upward. Erase all part left of $x=-1$. Can you do the same for other part?

PS: : The best method is to plot using one of many graphical calculators available and then copy them in you notebook of course, by hand.