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How do you find the range of #f(x)= (x^3+1)^-1#?

Answer 1

Julia Adams

#{y|y!=0}#

Determine the domain and range of the inverse function.

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

Flip #x# and #y#.

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

Now isolate #y#.

#x=(y^3+1)^-1# #x=1/(y^3+1)# #(y^3+1)x=(y^3+1)1/(y^3+1)# #(y^3+1)x=1# #y^3x+x=1# #y^3x+x-x=1-x# #y^3x=1-x# #(1/x)y^3x=(1/x)(1-x)# #y^3=(1-x)/x# #root(3)(y^3)=root(3)((1-x)/x)# #y=root(3)((1-x)/x)# #color(blue)(f^-1(x)=root(3)((1-x)/x))#

Now looking for the domain of the inverse function, we will find that it will only be undefined at #x=0#. So its domain is #{x|x!=0}#.

Therefore, the range of #f(x)=(x^3+1)^-1# is #{y|y!=0}#.

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Answer 2

Kennedy Adams

To find the range of the function ( f(x) = (x^3 + 1)^{-1} ), we consider the behavior of the function as ( x ) approaches positive and negative infinity. As ( x ) approaches positive infinity, ( x^3 ) dominates the expression, making ( (x^3 + 1)^{-1} ) approach zero. Similarly, as ( x ) approaches negative infinity, ( x^3 ) dominates the expression, making ( (x^3 + 1)^{-1} ) approach zero. Therefore, the range of the function is all real numbers except zero.

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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.