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

[mathdad]

Why must we restrict y = sqrt{anything}?

Is y = sqrt{anything} one-to-one?

 

[MarkFL]

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.

 

[mathdad]

Can you explain the difference between f(x) and g(x)?

 

[MarkFL]

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]

 

[mathdad]

I see they both pass the vertical line test.