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

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The discussion centers on whether the function y = sqrt{anything} qualifies as a one-to-one function, highlighting that it depends on the specific expression under the square root. The function f(x) = sqrt{x} is identified as one-to-one, while g(x) = sqrt{sin(x) + 1} is not, due to the periodic nature of the sine function affecting its output. The participants suggest plotting the graphs of both functions to visually compare their behaviors. Both functions pass the vertical line test, confirming they are functions, but their one-to-one status differs based on their definitions. Understanding these distinctions is crucial for determining the characteristics of such functions.
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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