Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f(x)=\frac{a x+b}{c x+d}\) and assume \(b c-a d \neq 0\). (a) Find the formula for \(f^{-1}(x)\). (b) Why is the condition \(b c-a d \neq 0\) needed? (c) What condition on \(a, b, c\), and \(d\) will make \(f=f^{-1}\) ?

Step-by-Step Solution

Answer

(a) \(f^{-1}(x) = \frac{b - xd}{cx - a}\). (b) \(bc-ad \neq 0\) ensures \(x\) can be solved. (c) \(a=d\) and \(b=-c\).

Step 1: Setup for Finding the Inverse

To find the inverse function \(f^{-1}(x)\), we need to start by setting \(y = f(x) = \frac{ax + b}{cx + d}\). We will solve for \(x\) in terms of \(y\).

Step 2: Cross-Multiply

Cross-multiply the equation \(y = \frac{ax + b}{cx + d}\) to obtain \(y(cx + d) = ax + b\). This gives \(ycx + yd = ax + b\).

Step 3: Rearrange Terms

Rearrange the terms to isolate \(x\): \(ycx - ax = b - yd\). This becomes \((yc - a)x = b - yd\).

Step 4: Solve for x

Solve for \(x\) by dividing both sides by \(yc - a\): \(x = \frac{b - yd}{yc - a}\). Thus the inverse function \(f^{-1}(x) = \frac{b - xd}{cx - a}\).

Step 5: Condition Analysis for Part (b)

The condition \(bc - ad eq 0\) ensures that \((yc - a) eq 0\) which is necessary for inversion as dividing by zero is undefined.

Step 6: Check Condition for f=f⁻¹ in Part (c)

For \(f = f^{-1}\), equate \(f(x)\) to \(f^{-1}(x)\): \(\frac{ax+b}{cx+d} = \frac{b - xd}{cx - a}\). Cross-multiplying gives \((ax+b)(cx-a) = (b-xd)(cx+d)\). After simplifying, find the conditions: for self-inversion, \(a=d\) and \(b=-c\).

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In the function \(f(x) = \frac{ax + b}{cx + d}\), the numerator \(ax + b\) and the denominator \(cx + d\) are linear polynomials.
These types of functions have interesting properties including vertical asymptotes, where the function is undefined. This usually happens when the denominator is zero.
In our example, \(f(x)\) is undefined when \(cx + d = 0\). Solving for \(x\) gives the vertical asymptote at \(x = -\frac{d}{c}\).
It's crucial to note that rational functions can be inverse functions only if a valid inverse exists, requiring a specific condition to be met to avoid division by zero. We'll explore these conditions in depth shortly.

Function Composition

Function composition involves applying one function to the results of another. It's noted by \((f \circ g)(x) = f(g(x))\).
With rational functions like \(f(x) = \frac{ax+b}{cx+d}\), composing the function with its inverse should ideally yield the identity function:\

• \(f(f^{-1}(x)) = x\)
• \(f^{-1}(f(x)) = x\)

For \(f(x)\) to have an inverse, it must be bijective (both injective and surjective).
This implies a one-to-one correspondence between the domain and range, ensuring that each \(x\) in the domain has a unique \(f(x)\), and every \(y\) in the range is achieved by some \(x\) in the domain.

Function Properties

Understanding the properties of rational functions helps us decipher how they behave and formulate their inverses. Each function has unique characteristics including:

• Domain: The set of all possible inputs (in \(f(x) = \frac{ax+b}{cx+d}\), all real numbers except where \(cx+d=0\))
• Range: The set of all possible outputs.
• Asymptotes: Lines that the graph approaches but never touches. Vertical asymptotes at \(x = -\frac{d}{c}\) and horizontal asymptotes often determined by the degree of the polynomials.

These elements must be considered when calculating the inverse \(f^{-1}(x)\), ensuring proper determination of the function's characteristics.

Self-Inverse Functions

A function that is its own inverse is called a self-inverse function. In simple terms, applying the function twice returns you to the original input, i.e., \(f(f(x)) = x\).
For a rational function \(f(x) = \frac{ax+b}{cx+d}\) to be self-inverse, it must satisfy specific conditions. Our step-by-step solution discovered that if \(a = d\) and \(b = -c\), then the function is self-inverse.
This means that under these conditions, \(f(x)\)'s formula rewrites in such a way that it behaves identically to its inverse, providing unique symmetry around the line \(y = x\). This special property can simplify solving equations or analyzing function behavior in certain contexts.