Does y = sqrt{anything} Qualify as a One-to-One Function?

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Discussion Overview

The discussion centers on whether the function defined as y = sqrt{anything} qualifies as a one-to-one function. Participants explore specific examples and the implications of different expressions under the square root, examining the conditions that affect one-to-one characteristics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that the one-to-one nature of y = sqrt{anything} depends on the specific expression inside the square root.
  • One example provided is f(x) = sqrt{x}, which is claimed to be one-to-one.
  • Another example, g(x) = sqrt{sin(x) + 1}, is proposed as not being one-to-one.
  • Participants express interest in visualizing the differences between f(x) and g(x) through graphing.
  • It is noted that both functions pass the vertical line test, which is a common method for determining if a function is well-defined.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the general question of whether y = sqrt{anything} is one-to-one, as the discussion highlights multiple competing views based on specific examples.

Contextual Notes

The discussion does not resolve the mathematical implications of the vertical line test in relation to one-to-one functions, nor does it clarify the conditions under which different expressions qualify as one-to-one.

mathdad
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Why must we restrict y = sqrt{anything}?

Is y = sqrt{anything} one-to-one?
 
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It depends on the "anything"...for example:

$$f(x)=\sqrt{x}$$

is one-to-one, while:

$$g(x)=\sqrt{\sin(x)+1}$$

is not one-to-one.
 
Can you explain the difference between f(x) and g(x)?
 
RTCNTC said:
Can you explain the difference between f(x) and g(x)?

Let's plot their graphs to see how they differ. :D

[DESMOS=-0.3404183173408105,19.659581682659205,-0.3390203078626133,6.59940120124406]y=\sqrt{x};y=\sqrt{1+\sin\left(x\right)}[/DESMOS]
 
I see they both pass the vertical line test.
 

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