MHB What is the range and inverse of the function y = 1/x?

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The function y = 1/x has a range of all real numbers except y = 0, which is confirmed by its graph showing curves in quadrants 1 and 3 that do not intersect the axes. The inverse of the function is also y = 1/x, with a domain of all real numbers except x = 0. This symmetry indicates that the range and domain are related, both excluding zero. The graph visually reinforces that the function approaches but never reaches the axes, validating the range. Understanding this behavior is crucial for novice math learners to grasp the concept of asymptotes in rational functions.
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Find the range algebraically.

y = 1/x

Find inverse of y.

x = 1/y

Solve for y.

yx = (1/y)(y)

yx = 1

y = 1/x

f^(-1) x = 1/x

The domain of f^(-1) x is ALL REAL NUMBERS except that x cannot be 0.

So, the range of y = 1/x is ALL REAL NUMBERS except that y cannot be 0.

Correct?
 
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$y = \dfrac{1}{x}$ is a basic parent function whose graph is easily sketched ... so, sketch it and answer your own question.
 
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?
 
you asked the same question in your other post ...

http://mathhelpboards.com/pre-calculus-21/range-functions-4-a-21775.html#post98495
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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