Problem 55 - A Graphical Approach to Precalculus with Limits

Question

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=3 x-4$$

Step-by-Step Solution

Answer

The inverse function is \(y = \frac{x+4}{3}\). Domain and range of both functions are all real numbers.

Step 1: Determine if the Function is One-to-One

A function is one-to-one if each output value is the result of exactly one input value. Since the function \(y = 3x - 4\) is linear with a non-zero slope (3 in this case), it must be one-to-one. This is because linear functions of the form \(y = mx + b\) (where \(m eq 0\)) are one-to-one.

Step 2: Express the Function in Terms of x

To find the inverse, start by expressing \(x\) in terms of \(y\) in the original function equation. Begin with the equation \(y = 3x - 4\). Add 4 to both sides to get \(y + 4 = 3x\). Then, divide both sides by 3 to solve for \(x\): \(x = \frac{y + 4}{3}\).

Step 3: Swap Variables to Find the Inverse

For the inverse function, swap \(x\) and \(y\) so that the equation becomes \(y = \frac{x + 4}{3}\). This new equation represents the inverse function \(f^{-1}(x)\).

Step 4: Identify the Domain and Range

For the original function \(f(x) = 3x - 4\), the domain is all real numbers \((-fty, fty)\) because any real number can be input into a linear function. The range is also all real numbers because linear functions can produce any real value. The inverse function \(f^{-1}(x) = \frac{x + 4}{3}\) similarly has a domain and range of all real numbers \((-fty, fty)\).

Step 5: Graph the Functions

Graph the original function \(f(x) = 3x - 4\), which is a straight line with slope 3 and y-intercept -4. Next, graph the inverse function \(f^{-1}(x) = \frac{x + 4}{3}\), which is also a straight line, but with slope \(\frac{1}{3}\) and y-intercept \(\frac{4}{3}\). The graphs should reflect each other over the line \(y = x\).

Key Concepts

One-to-One Functions

One-to-one functions are special because they have a unique characteristic: every input value is paired with one distinct output value, and no output is repeated for different inputs. This means that each element in the domain of the function maps to a unique element in the range. If a function passes the

• Horizontal Line Test

where any horizontal line drawn through the graph intersects the curve at most once, then the function is one-to-one.

For instance, the function given, \(y = 3x - 4\), is one-to-one because it is a linear function with a non-zero slope. Linear functions of the form \(y = mx + b\) (where \(m eq 0\)) are always one-to-one because such functions increase or decrease steadily, never producing the same output value twice. This is essential for inverse functions, as only one-to-one functions have inverses that are also functions.

Linear Functions

Linear functions are among the simplest types of functions, defined by expressions of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. These functions graph as straight lines and are characterized by a constant rate of change, which is the value of the slope \(m\).

In the function \(y = 3x - 4\), the slope \(m\) is 3, meaning that for each unit increase in \(x\), \(y\) increases by 3 units. The y-intercept \(b\) is -4, indicating that when \(x = 0\), \(y\) will be -4. The line crosses the x-axis and y-axis at these intercept points.

• Constant slope
• Straight line graph

Linear functions provide a straightforward way to understand and manipulate algebraic expressions, making them a cornerstone in algebra and calculus.

Domain and Range

Understanding the domain and range of a function is critical for grasping its behavior and limitations. The domain of a function is the complete set of possible input values (\(x\) values), while the range is the complete set of possible output values (\(y\) values) the function can produce.

For the function \(f(x) = 3x - 4\), both the domain and the range are all real numbers, or \((-\infty, \infty)\). This is because linear functions have no restrictions on input; any real number can be used for \(x\). Consequently, any real number can be produced as an output.

• Domain: All real numbers
• Range: All real numbers

Similarly, its inverse \(f^{-1}(x) = \frac{x + 4}{3}\) also has a domain and a range of all real numbers. Thus, the inverse maintains the same expansive domain and range as the original function.

Graphing Functions

Graphing functions helps visualize their behavior and understand their properties better. To graph a linear function like \(f(x) = 3x - 4\), one can start by plotting the y-intercept \((-4)\), then use the slope \(3\) to find other points by rising \(3\) units for every unit moved to the right.

For its inverse \(f^{-1}(x) = \frac{x + 4}{3}\), start by plotting the y-intercept \((\frac{4}{3})\), and use the slope \(\frac{1}{3}\) to find other points. Each of these functions will graph as a straight line.

• Original function: Slope 3, y-intercept -4
• Inverse function: Slope \(\frac{1}{3}\), y-intercept \(\frac{4}{3}\)

When graphed together, these lines should be symmetrical over the line \(y = x\), demonstrating how the inverse function "undoes" the original function's operations, reflecting the concept of symmetry between a function and its inverse.