Question

Group Activity Inverse Functions Let \(f(x)=\frac{a x+b}{c x+d}, \quad c \neq 0, \quad a d-b c \neq 0\) (a) Writing to Learn Give a convincing argument that \(f\) is one-to-one. (b) Find a formula for the inverse of \(f\) . (c) Find the horizontal and vertical asymptotes of \(f\) . (d) Find the horizontal and vertical asymptotes of \(f^{-1} .\) How are they related to those of \(f ?\)

Short answer

The function \(f\) is indeed one-to-one. Its inverse is given by \(f^{-1}(x) = \frac{dx - b}{-cx + a}\). The vertical asymptote of \(f\) is \(x = -d/c\) and the horizontal asymptote is \(y = a/c\). For the inverse function \(f^{-1}\), the vertical asymptote is \(x = a/c\) and the horizontal asymptote is \(y = -d/c\). The vertical asymptote of \(f\) is the horizontal asymptote of \(f^{-1}\) and vice versa.

Step-by-step solution

Proving one-to-one property

To prove that the function \(f\) is one-to-one, assume that \(f(x_1) = f(x_2)\) for any \(x_1\) and \(x_2\). This would mean \(\frac {ax_1+b}{cx_1+d} = \(\frac{ax_2+b}{cx_2+d}\). Cross multiplying and simplifying, we get \(ad(x_1 - x_2) = bc(x_1 - x_2)\). Given that \(ad - bc \neq 0\), the only way the equation can hold true is if \(x_1 = x_2\). Thus, \(f\) is indeed a one-to-one function.

Finding the inverse of f

The inverse of the function \(f\) can be found by interchanging \(x\) and \(y\) in \(f(y) = \(\frac{ay+b}{cy+d}\), then solve for \(y\). The result is \(f^{-1}(x) = \frac{dx - b}{-cx + a}\) which can be confirmed by checking that \(f(f^{-1}(x)) = x\).

Finding the asymptotes of f

For the given function \(f\), the vertical asymptote is \(x = -d/c\) which can be found by determining the \(x\) for which \(f\) is undefined. The horizontal asymptote, which exists when \(x\) approaches infinity, is at \(y = a/c\) for the function \(f\).

Finding the asymptotes of inverse f

For the inverse function \(f^{-1}\), the vertical asymptote is \(x = a/c\) and the horizontal asymptote is \(y = -d/c\). These are obtained again by determining where the function is undefined and where \(y\) goes to as \(x\) approaches infinity.

Relationship between asymptotes of f and its inverse

Looking at the asymptotes of the function \(f\) and its inverse \(f^{-1}\), we can observe that the vertical asymptote of \(f\) is the horizontal asymptote of \(f^{-1}\) and vice versa. This is because the function \(f\) and its inverse \(f^{-1}\) are reflections of each other over the line \(y = x\).