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How do you find the inverse of #f(x) = x^2 +7#?

Answer 1

Sophie Adams

It depends on the domain A.
#f : A rightarrow mathbb{R}#

Exchange x and f(x) = y.

#x = y^2 + 7#

#x - 7 = y^2#

#±sqrt{x-7} = f^-1(x)#

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Answer 2

Chloe Alexander

To find the inverse of the function ( f(x) = x^2 + 7 ), follow these steps:

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

Starting with ( f(x) = x^2 + 7 ):

Replace ( f(x) ) with ( y ): ( y = x^2 + 7 ).
Swap the variables: ( x = y^2 + 7 ).
Solve for ( y ): ( x - 7 = y^2 ). Taking the square root of both sides (considering both positive and negative roots): ( y = \sqrt{x - 7} ) or ( y = -\sqrt{x - 7} ).
Replace ( y ) with ( f^{-1}(x) ): ( f^{-1}(x) = \sqrt{x - 7} ) or ( f^{-1}(x) = -\sqrt{x - 7} ).

So, the inverse of ( f(x) = x^2 + 7 ) is ( f^{-1}(x) = \sqrt{x - 7} ) or ( f^{-1}(x) = -\sqrt{x - 7} ).

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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.