MHB How can I find the range of a hyperbolic curve using its graph?

[mathdad]

Find the range using the graph of y.

y = 1/x

This function is weird. It has a curve in quadrants 1 and 3 that does not cross the lines y = 0 and x = 0.

How can I determine the range of such a graph?

 

[skeeter]

domain is $x \in \mathbb{R}, \, x \ne 0$

range is $y \in \mathbb{R}, \, y \ne 0$

why so difficult to interpret the graph?[DESMOS=-10.10016694490818,9.89983305509182,-10.45,9.55]y=\frac{1}{x}[/DESMOS]

 

[mathdad]

I know what this graph looks like. I've seen it hundreds of times but what does it mean to a novice math learner? There is a curve in quadrants 1 and 3 that does not cross the lines x = 0 and y = 0. The textbook answer is (-infinity, 0) U (0, infinity). What on the graph tells me that this is the correct range?

 

[skeeter]

There is a curve in quadrants 1 and 3 that does not cross the lines x = 0 and y = 0.

if a graph of a function does not cross a vertical line like x = 0, then that line is a vertical asymptote ... x = 0 is excluded from the function's domain.

if a graph of a function does not cross a horizontal line like y = 0, then that line is a horizontal asymptote ... y = 0 is excluded from the function's range.

can't put it any plainer than that ... maybe you should select video(s) from the link for alternative, non-textbook explanations

https://www.google.com/search?q=ide..._eHUAhUY6GMKHSHzDhAQ_AUICigB&biw=1366&bih=638

 

[mathdad]

I'll seek more video help than textbooks. I can find just about anything on youtube.com. I will use this site when a video clip makes no sense.

 

[MarkFL]

I know what this graph looks like. I've seen it hundreds of times but what does it mean to a novice math learner? There is a curve in quadrants 1 and 3 that does not cross the lines x = 0 and y = 0. The textbook answer is (-infinity, 0) U (0, infinity). What on the graph tells me that this is the correct range?

It may interest you to know that the curve:

$$y=\frac{1}{x}$$

is actually a hyperbolic curve. Consider the graph of the hyperbolic curve:

$$x^2-y^2=1$$

and its asymptotes, given by:

$$x^2=y^2$$

[DESMOS=-5,5,-1.7346053772766694,1.7346053772766694]x^2-y^2=1;x^2=y^2[/DESMOS]

It can be shown that by rotating our axes by $$\frac{\pi}{4}$$, the graphed hyperbola becomes:

$$y=\frac{1}{x}$$

I'll wait until you get to the section on rotation of axes before we explore that further. :D

 

[mathdad]

It may interest you to know that the curve:

$$y=\frac{1}{x}$$

is actually a hyperbolic curve. Consider the graph of the hyperbolic curve:

$$x^2-y^2=1$$

and its asymptotes, given by:

$$x^2=y^2$$
It can be shown that by rotating our axes by $$\frac{\pi}{4}$$, the graphed hyperbole becomes:

$$y=\frac{1}{x}$$

I'll wait until you get to the section on rotation of axes before we explore that further. :D

Cool. Continue to answer my questions. I appreciate your guidance through my review of a great course. I will use youtube precalculus clips to review each chapter and post questions here when needed. Check your inbox.