Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-24: Determine the domain and range of \(f^{-1}\) for the given function without actually finding \(f^{-1}\). Hint: First find the domain and range of \(f\). $$ f(x)=\frac{4 x+5}{3 x-8} $$

Step-by-Step Solution

Answer

Domain of \(f^{-1}\): all real numbers except \(\frac{4}{3}\); Range of \(f^{-1}\): all real numbers except \(\frac{8}{3}\).

Step 1: Understanding the problem

We need to determine the domain and range of the inverse function \(f^{-1}\) by first finding the domain and range of the given function \(f(x) = \frac{4x + 5}{3x - 8}\).

Step 2: Determine the Domain of \(f\)

To find the domain of \(f(x)\), observe where the function is undefined. This happens when the denominator is zero. Solve \(3x - 8 = 0\), which gives \(x = \frac{8}{3}\). Therefore, the domain of \(f\) is all real numbers except \(x = \frac{8}{3}\).

Step 3: Determine the Range of \(f\)

To find the range of \(f\), look for the values \(y\) that \(f(x)\) cannot take. Since \(f(x)\) is a rational function, it cannot equal its horizontal asymptote. The horizontal asymptote for \(\frac{4x + 5}{3x - 8}\) is found by evaluating the ratio of the leading coefficients, \(\frac{4}{3}\). Thus, the range of \(f\) is all real numbers except \(y = \frac{4}{3}\).

Step 4: Determine the Domain of \(f^{-1}\)

The domain of an inverse function \(f^{-1}\) is the range of \(f\). From the range of \(f\), the domain of \(f^{-1}\) is all real numbers except \(\frac{4}{3}\).

Step 5: Determine the Range of \(f^{-1}\)

The range of an inverse function \(f^{-1}\) is the domain of \(f\). From the domain of \(f\), the range of \(f^{-1}\) is all real numbers except \(\frac{8}{3}\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (typically represented as "x") for which the function is defined. To determine the domain of a rational function, such as \(f(x) = \frac{4x + 5}{3x - 8}\), we need to check where the function becomes undefined. This usually happens when the denominator equals zero, leading to division by zero, which is not allowed. To find where the function \(f(x)\) is undefined, set the denominator equal to zero:\[3x - 8 = 0\]Solving this equation, we find:\[x = \frac{8}{3}\]Hence, the domain consists of all real numbers except \(x = \frac{8}{3}\). This means you can plug any real number into the function except \(\frac{8}{3}\).

• The function is undefined at \(x = \frac{8}{3}\).
• The domain is all real numbers except this value: \(x eq \frac{8}{3}\).

Range of a Function

The range of a function is the set of all possible output values (typically represented as "y") that a function can take. For rational functions, the range can often be found by identifying the horizontal asymptotes, which represents a value that the function approaches but never actually reaches. In the case of \(f(x) = \frac{4x + 5}{3x - 8}\), we find the horizontal asymptote by examining the ratio of the leading coefficients in the function, because as \(x\) tends towards infinity, those terms dominate the behavior of the function. Here, the horizontal asymptote is\[y = \frac{4}{3}\]As such, the range of \(f(x)\) is all real numbers except \(y = \frac{4}{3}\). This means \(f(x)\) can produce any real number apart from \(\frac{4}{3}\).

• The horizontal asymptote limits the range.
• \(y\) cannot equal \(\frac{4}{3}\).

Rational Functions

Rational functions are functions represented by the ratio of two polynomials. The general form of a rational function is \(f(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\). A key characteristic of rational functions is that they can have asymptotes:

• Vertical asymptotes occur where the function becomes undefined (where \(Q(x) = 0\)).
• Horizontal asymptotes depend on the degree and leading coefficients of the numerator and denominator polynomials.

In our exercise, \(f(x) = \frac{4x + 5}{3x - 8}\), the vertical asymptote is at \(x = \frac{8}{3}\) since this makes the denominator zero. Meanwhile, the horizontal asymptote is determined by the leading coefficients, \(4/3\), indicating that as \(x\) becomes very large or very negative, the function approaches \(y = \frac{4}{3}\).Understanding such properties helps with determining domains, ranges, and behaviors of rational functions.