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

Answer 1

Liam Brennan

Write #y= 2x^2 +3#. Exchange x , y and solve for y.

x=# 2y^2 +3#

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

y = #sqrt((x-3)/2)# is the required inverse function of f(x)

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Answer 2

Xavier Alonso

To find the inverse function of ( f(x) = 2x^2 + 3 ), we can follow these steps:

Replace ( f(x) ) with ( y ): ( y = 2x^2 + 3 ).
Swap ( x ) and ( y ): ( x = 2y^2 + 3 ).
Solve for ( y ) in terms of ( x ).

[ \begin{aligned} x &= 2y^2 + 3 \ 2y^2 &= x - 3 \ y^2 &= \frac{x - 3}{2} \ y &= \pm \sqrt{\frac{x - 3}{2}} \end{aligned} ]

Since a function can only have one output for each input, we take the positive square root to get the inverse function:

[ f^{-1}(x) = \sqrt{\frac{x - 3}{2}} ]

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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.