A Function and Its Inverse - for nycmath

[Math Tutor]

Let [tex]f(x)=\dfrac{2x-3}{x+1}[/tex].

(a) Give the domain and the range of [tex]f[/tex].
(b) Show that [tex]f[/tex] is one-to-one and find [tex]f^{-1}(x)[/tex].
(c) Compare the domain and range of [tex]f^{-1}[/tex] with those of [tex]f[/tex]. What do you notice?
(d) Solve [tex]f(x)=f^{-1}(x)[/tex].

Part (d) is the interesting one. Most students start by setting [tex]f(x)=x[/tex], on the idea that the two graphs can only meet on the line [tex]y=x[/tex]. Try it the direct way instead and see what the algebra tells you about where the two curves cross.