How do you graph #f(x)=3/x# and then use the horizontal test to determine whether the inverse of f is a function?

1 Answer
MathFact-orials.blogspot.com
Jul 7, 2017

See below:

Explanation:

Let's first graph #f(x)=3/x# .

It's sometimes easier to see and work with if we say that #y=f(x)#, giving:

#y=3/x#

We can plot a few points, such as:

#((x,y),(3,1),(-3,-1),(9,1/3),(-9,-1/3),(0,"undefined"))#

and it graphs as:

graph{3/x}

Inverse

Now let's find the inverse of the function. To do that, we switch #x# for #y# and vice versa, then solve for #y#:

#x=3/y#

#xy=3#

#y=3/x#

Is #y=3/x# a function?

We can use the vertical line test (for any given test line dropped vertically, it will intersect our graph in no more than one place) to show that this is indeed a function. I'll drop a couple of test lines to show:

graph{(y-3/x)(x-0y-2)(x-0y+3)=0}

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