Problem 56 - 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. $$y=4 x-5$$

Step-by-Step Solution

Answer

The inverse function is \(y = \frac{x+5}{4}\); both functions have domain and range \((- fty, fty)\).

Step 1: Verify One-to-One Property

A function is one-to-one if each input gives a unique output. The function given is \(y = 4x - 5\), which is a linear function with a non-zero slope. Linear functions with non-zero slopes are always one-to-one because they have a constant rate of change and do not repeat values.

Step 2: Swap and Solve for the Inverse

For a one-to-one function \(y = f(x) = 4x - 5\), find the inverse by swapping \(x\) and \(y\) and then solving for \(y\). Begin with \(x = 4y - 5\). Add 5 to both sides to get \(x + 5 = 4y\), then divide by 4 to get \(y = \frac{x + 5}{4}\). Thus, the inverse function is \(y = f^{-1}(x) = \frac{x+5}{4}\).

Step 3: Domain and Range Analysis

The original function \(f(x) = 4x - 5\) is defined for all real numbers, hence its domain is \((-fty, fty)\). As a linear function, its range is also \((-fty, fty)\). The inverse function \(f^{-1}(x) = \frac{x+5}{4}\) is also a linear function with domain and range \((-fty, fty)\).

Step 4: Graphing Instructions

To graph the function \(f(x) = 4x - 5\) and its inverse \(f^{-1}(x) = \frac{x + 5}{4}\), plot them on the same axes. The function \(f(x)\) is a line with slope 4 and y-intercept at -5. The inverse \(f^{-1}(x)\) is a line with slope \(\frac{1}{4}\) and y-intercept at \(\frac{5}{4}\). Both graphs should appear symmetric with respect to the line \(y = x\).

Step 5: Confirm Inverses by Symmetry

To ensure the functions \(f(x)\) and \(f^{-1}(x)\) are inverses, check that they are reflections over the line \(y = x\). Plot this line for visual confirmation: \( (x, f(x)) \) correspond to \( (f^{-1}(x), x) \) on the graph, indicating symmetry.

Key Concepts

One-to-One Function

In mathematics, a one-to-one function, also known as an injective function, is a function in which every element of the domain is mapped to a unique element in the codomain. This means that no two different inputs will produce the same output.
In simpler terms, a one-to-one function does not repeat any y-values for different x-values.

• For example, the function \(y = 4x - 5\) is a one-to-one function because for every value of \(x\), there is a unique value of \(y\).
• One method to check if a function is one-to-one is by performing the horizontal line test on its graph. If no horizontal line intersects the graph more than once, the function is one-to-one.

Linear functions (like our example \(y = 4x - 5\)) with non-zero slopes are always one-to-one, as their constant rate of change ensures no repetition of values over the domain.

Domain and Range

Understanding the domain and range of functions is fundamental in calculus and algebra. The domain of a function is the complete set of all possible input values (\(x\)) that the function can accept, while the range is the complete set of all possible output values (\(y\)) that the function can produce.

• For the function \(f(x) = 4x - 5\), its domain includes all real numbers, \((-\infty, \infty)\), because any real number can be substituted for \(x\).
• The range is also all real numbers, \((-\infty, \infty)\), as any real number can be yielded by the function as we vary \(x\).
• Similarly, the inverse function \(f^{-1}(x) = \frac{x+5}{4}\) has the same domain and range: both are all real numbers, illustrating a symmetrical characteristic of inverses.

Inverse functions essentially swap the domain and range of the original function. This symmetry emphasizes the importance of correctly identifying both sets for accurate mathematical analysis.

Graphing Functions

Graphing functions, along with their inverses, provides a clearer visual understanding of their relationships and behaviors.

• To graph the function \(y = 4x - 5\), plot the y-intercept at \(-5\), and use the slope of \(4\) to rise four units for every one unit moved right or left.
• For its inverse \(y = f^{-1}(x) = \frac{x+5}{4}\), plot the y-intercept at \(\frac{5}{4}\), and proceed with a gentler slope of \(\frac{1}{4}\).

Both graphs should exhibit symmetry with respect to the line \(y = x\). This line serves as a mirror, reflecting each point \((x, y)\) from the original function to the point \((y, x)\) on the inverse. Visual confirmation of this symmetry reinforces the inverse relationship.
This method not only strengthens comprehension of the function itself but also solidifies the idea of inverse functions as reflections across the line \(y = x\).