The given function is
[tex]f(x)=\frac{4}{x^3-1}[/tex]
The inverse function is
[tex]f^{-1}(x)=\sqrt[3]{\frac{4}{x}+1}[/tex]
Recall that the range of the function is the completely possible set of resulting values of the function.
Consider the function
[tex]f(x)=\frac{4}{x^3-1}[/tex]
This is given function rational polynomial function so the range is negative infinity to positive infinity except for the value of x where the denominator is zero.
[tex]x^3-1=0[/tex][tex]x^3=1[/tex][tex]x=1[/tex]
Hence the given function is not valid at x=1.
The range of the function is
[tex](-\infty,\infty)-\mleft\lbrace1\mright\rbrace[/tex]
Consider the inverse function
[tex]f^{-1}(x)=\sqrt[3]{\frac{4}{x}+1}[/tex]
This is inverse function rational polynomial function so the range is negative infinity to positive infinity except for the value of x where the denominator is zero.
[tex]\sqrt[3]{x}=0[/tex][tex]x=0[/tex]
Hence the inverse function is not valid at x=0.
The range of the inverse function is
[tex](-\infty,\infty)-\mleft\lbrace0\mright\rbrace[/tex]
