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What is the inverse function of #f(x)=11-3x^2#?

Answer 1

Sadie Alexander

The function is not one-to-one. It does not have an inverse.

From #y = 11-3x^2#, if we try to solve for #x# int terms of #y#, we get:

#x^2 = (11-y)/3#.

But for every value of #y# except #y=11#, there are two possible values of #x#, namely #+-sqrt ((11-y)/3)#.

Functions never return 2 values. So #x# is not a function of #y#, and #f(x)# is not invertible.

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Answer 2

Chloe Alexander

To find the inverse function of ( f(x) = 11 - 3x^2 ), follow these steps:

Replace ( f(x) ) with ( y ).
Swap ( x ) and ( y ).
Solve for ( y ).
Replace ( y ) with ( f^{-1}(x) ).

[ y = 11 - 3x^2 ] [ x = 11 - 3y^2 ] [ 3y^2 = 11 - x ] [ y^2 = \frac{11 - x}{3} ] [ y = \pm \sqrt{\frac{11 - x}{3}} ]

Thus, the inverse function of ( f(x) = 11 - 3x^2 ) is: [ f^{-1}(x) = \pm \sqrt{\frac{11 - x}{3}} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.