Injective Function - 1.9 #5

[Guganesan S. Ilavarasan]

Can anyone please tell me how to find whether an equation is injective?

Let x∈R. Which of the following functions is(are) injective?

f(x)=√10−x

f(x)=7x+63x

f(x)=2x+9

f(x)=(5x+4)(2x−3)2

And why the 4th option is not injective?

[Akshay Malik]

[Akshay Malik]

[Anand Iyer]

The first three are injective, since, for no two x-values, the y-value will be the same.  2 and 3 are linear functions, and the first one is treated as linear due to limitations imposed on square-root functions (we consider only positive square-roots)

However, note that the last one is a quadratic function.  This is a parabola that opens up.  If you draw this parabola, you can see that y-value will be the same for x-values, irrespective of its signage.

In this particular case, have a look at the figure...you'll find that same y-value occurs for two x-values - one on either side of the parabola.

[Anand Iyer]

Well, on the first option, it's not exactly linear, but we look at only one half of the parabola.

[Guganesan S. Ilavarasan]

Thank you for your efforts gentlemen, but Mr Anand, please forgive my ignorance as I have been away from quadratic equations for a long time. But how exactly you found out by , merely looking at the equation as it's a linear or a parabolic, sir?  And by logic all Parabola are Non - Injective isn't it, sir?

[Malabika Guha Mustafi]

@ Guganesan,

 by seeing the x exponential power.

If x power is 1( y=mx +c) then linear if it is square then  parabolic.(ax^2+bx +c)

Is n't It Anand?

[Guganesan S. Ilavarasan]

But there isn't any exponential in the given option madam. It's divided by 2. Option No: 4

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Guganesan Ilavarasan University of Madras | M.A. 2018-20 Department of Ancient History and Archaeology Primary | +91-9176531434 (WhatsApp & Telegram)

[Malabika Guha Mustafi]

:) @ Guganesan , What is your opinion about the numerator part? Product of two x is square of x.

[Anand Iyer]

yes, that's right Malabika.  square functions are parabolic.

However, you can't say there are no quadratic functions.  Some might be, It depends on the domain.  If the domain isn't explicitly provided, we typically consider all reals.  In that case, yes, no quadratic functions are injective  

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Cheers,

Anand

[Anand Iyer]

The second sentence should be

However, you can't say there are no injective quadratic functions.

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Cheers,

Anand

[Malabika Guha Mustafi]

The first option is actually a quadratic equation and injective just presentation is different.