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How do you find the inverse of #f(x)=sqrt(x+1) + 3# and is it a function?

Answer 1

Serenity Jones

The inverse of a function can be found algebraically by switching the x and y values.

#y = sqrt(x + 1) + 3#

#x = sqrt(y + 1) + 3#

#x - 3 = sqrt(y + 1)#

#(x - 3)^2 = (sqrt(y + 1))^2#

#x^2 - 6x + 9 = y + 1#

#x^2 - 6x + 8 = y#

This is a quadratic, therefore it is a function.

Hopefully this helps!

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Answer 2

Madelyn Anderson

To find the inverse of the function ( f(x) = \sqrt{x + 1} + 3 ), follow these steps:

Replace ( f(x) ) with ( y ).
Swap the roles of ( x ) and ( y ): ( x = \sqrt{y + 1} + 3 ).
Solve for ( y ).
Square both sides to eliminate the square root: ( (x - 3)^2 = y + 1 ).
Subtract 1 from both sides: ( y = (x - 3)^2 - 1 ).

The inverse function is ( f^{-1}(x) = (x - 3)^2 - 1 ).

Yes, the inverse function is a function because each input value ( x ) corresponds to exactly one output value ( y ), satisfying the vertical line test. Therefore, both the original function and its inverse are functions.

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Answer from HIX Tutor

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