Chapter 4 - Inverse, Exponential, and Logarithmic Functions - 4.1 Inverse Functions - 4.1 Exercises - Page 417: 46

Yes.

Use the definition of inverses, for $f(x)=\frac{x-3}{x+4}$ and $g(x)=\frac{4x+3}{1-x}$, we have: $f(g(x))=\frac{(\frac{4x+3}{1-x})-3}{(\frac{4x+3}{1-x})+4}=\frac{4x+3-3+3x}{4x+3+4-4x}=\frac{7x}{7}=x$ $g(f(x))=\frac{4(\frac{x-3}{x+4})+3}{1-(\frac{x-3}{x+4})}=\frac{4x-12+3x+12}{x+4-x+3}=\frac{7x}{7}=x$ Thus f and g are inverses to each other.