Problem 23 - Algebra 2

Question

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=3 x+4 $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x) = 3x + 4\) is \(f^{-1}(x) = \frac{1}{3}x - \frac{4}{3}\). The domain and range of both \(f\) and \(f^{-1}\) are all real numbers \(-\infty < x < \infty\) and \(-\infty < y < \infty\). \(f^{-1}\) is a function.

Step 1: Find the Inverse of \(f(x)\)

To find the inverse of the function \(f(x) = 3x + 4\), first, replace \(f(x)\) with \(y\). This gives us \(y = 3x + 4\). Now, swap \(y\) and \(x\); this means we have \(x = 3y + 4\). Solve the equation for \(y\) to get the inverse function. In this case, first subtract 4 from both sides to obtain \(x - 4 = 3y\). Then divide both sides by 3 to finish isolating \(y\) - this gives us \(y = \frac{1}{3}x - \frac{4}{3}\). Therefore, \(f^{-1}(x) = \frac{1}{3}x - \frac{4}{3}\).

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

For the linear function \(f(x) = 3x + 4\), the domain is all real numbers, i.e., \(-\infty < x < \infty\), because there's no limitation on \(x\). The range is also all real numbers \(-\infty < y < \infty\) because there aren't any restrictions placed on \(y\) either.

Step 3: Determine the Domain and Range of \(f^{-1}\)

Looking at the inverse function \(f^{-1}(x) = \frac{1}{3}x - \frac{4}{3}\), like \(f\), the domain and range are both all real numbers, \(-\infty < x < \infty\) and \(-\infty < y < \infty\). This is because the inverse function is also linear and unrestricted.

Step 4: Determine Whether \(f^{-1}\) is a Function

A relation or operation is considered a function if and only if each input has exactly one output. In this case, each input \(x\) in \(f^{-1}(x) = \frac{1}{3}x - \frac{4}{3}\) yields precisely one output value of \(y\), so \(f^{-1}\) is indeed a function.

Key Concepts

Linear Functions

Linear functions are one of the most basic types of functions in mathematics. They are described by equations of the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. In the exercise provided, our linear function is \(f(x) = 3x + 4\). The key characteristics of linear functions include:

• They graph as straight lines.
• The slope of the line is determined by the coefficient \(a\) (3 in our exercise), which indicates the steepness or incline of the line.
• The y-intercept is the value of \(b\) (4 in our exercise), indicating where the line crosses the y-axis.

Linear functions are widely used due to their simplicity and the straightforward nature of their graphs. They have no breaks or curves, making them predictable and easy to understand.

Domain and Range

The domain and range of a function describe the set of possible input and output values, respectively. For linear functions like \(f(x) = 3x + 4\), both the domain and range are all real numbers. This means:

• The domain \(\(-\infty, \infty\)\) suggests there is no restriction on what \(x\) values can be to identify \(f(x)\).
• The range \(\(-\infty, \infty\)\) indicates that any \(y\) value can be achieved as the function extends infinitely in both the positive and negative directions.

For the inverse function \(f^{-1}(x) = \frac{1}{3}x - \frac{4}{3}\), the domain and range are also all real numbers. This is a property of linear functions in general: their inverses remain functionally unrestricted in precisely the same manner as the original function. So, you can think of the domain as the set of all possible inputs, and the range as the set of all achievable outputs.

Function Properties

Understanding function properties is crucial in determining whether an inverse is a function. A function is defined as a relation where each input corresponds to exactly one output. This is easy to verify with linear functions using the vertical line test:

• If a vertical line intersects the graph of a function at more than one point, it suggests that not every input has a unique output, and therefore, it is not a function.
• Our exercise confirms that \(f(x) = 3x + 4\) and its inverse \(f^{-1}(x) = \frac{1}{3}x - \frac{4}{3}\) both pass the vertical line test, hence they are functions.

Linear functions and their inverses hold a key feature in mathematics: they consistently produce a unique value for every input, confirming their status as functions. This exclusivity of input-output sets is what makes functions so robust and reliable.