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 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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