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How do you find the inverse of the function: #f(x)= 6+ sqrt(x+7)#?

Answer 1

Caleb Alexander

Inverse of #f(x)=6+sqrt(x+7)# is #g(x)=(x-6)^2-7#

Let #y=f(x)=6+sqrt(x+7)#

then #sqrt(x+7)=y-6#

and #x+7=(y-6)^2#

and #x=(y-6)^2-7#

Hence, inverse of #f(x)=6+sqrt(x+7)# is #g(x)=(x-6)^2-7#

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Answer 2

Sophie Adams

To find the inverse of the function ( f(x) = 6 + \sqrt{x+7} ), follow these steps:

Start with the given function ( f(x) ).
Replace ( f(x) ) with ( y ).
Swap ( x ) and ( y ) to interchange the roles of ( x ) and ( y ). The equation will become ( x = 6 + \sqrt{y + 7} ).
Solve for ( y ).
Begin by isolating the square root term by subtracting 6 from both sides: ( x - 6 = \sqrt{y + 7} ).
To eliminate the square root, square both sides of the equation: ( (x - 6)^2 = (\sqrt{y + 7})^2 ).
Simplify the equation: ( (x - 6)^2 = y + 7 ).
Next, isolate ( y ) by subtracting 7 from both sides: ( (x - 6)^2 - 7 = y ).
Finally, replace ( y ) with ( f^{-1}(x) ) to represent the inverse function. Therefore, the inverse function is ( f^{-1}(x) = (x - 6)^2 - 7 ).

So, the inverse function of ( f(x) = 6 + \sqrt{x+7} ) is ( f^{-1}(x) = (x - 6)^2 - 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.