The function given is,
[tex]f(x)=\frac{4}{7+x}[/tex]
Let us now find the inverse of the above function
[tex]\mathrm{A\: function\: g\: is\: the\: inverse\: of\: function\: f\: if\: for}\: y=f\mleft(x\mright),\: \: x=g\mleft(y\mright)[/tex]
Replace x with y
[tex]x=\frac{4}{7+y}[/tex]
Solve for y
[tex]\begin{gathered} x(7+y)=4 \\ \frac{x(7+y)}{x}=\frac{4}{x} \\ 7+y=\frac{4}{x} \\ \therefore y=\frac{4}{x}-7 \end{gathered}[/tex]
Therefore,
[tex]f^{-1}(x)=\frac{4}{x}-7[/tex]
Domain of f(x) is
[tex]\lbrace x|x\ne-7\rbrace\text{ Option B}[/tex]
Range of f(x) is
[tex]\lbrace y|y\ne0\rbrace\text{ Option B}[/tex]
Domain of the inverse function
[tex]\lbrace x|x\ne0\rbrace\text{ Option C}[/tex]
Range of the inverse function
[tex]\lbrace y|y\ne-7\rbrace\text{ Option C}[/tex]
