Problem 56 - A Graphical Approach to College Algebra

Question

For each function that is one-to-one, write an equation for the inverse function 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 is \(y = \frac{x+5}{4}\). Domain and range are all real numbers.

Step 1: Check if the function is one-to-one

A function is one-to-one if each value of the function corresponds to exactly one value of the input. The function \(y = 4x - 5\) is a linear function with a non-zero slope, making it one-to-one since it is strictly increasing.

Step 2: Interchange x and y

Starting with the equation \(y = 4x - 5\), swap the variables to find its inverse: \(x = 4y - 5\).

Step 3: Solve for y

Solve the equation \(x = 4y - 5\) for \(y\). Add 5 to both sides to get \(x + 5 = 4y\), then divide both sides by 4: \(y = \frac{x + 5}{4}\). This is the inverse function \(y = f^{-1}(x)\).

Step 4: Graph the functions

Graph the original function \(y = 4x - 5\) and its inverse \(y = \frac{x + 5}{4}\). The graph of the inverse is a reflection of the original function across the line \(y = x\).

Step 5: Determine the domain and range

For \(f(x) = 4x - 5\), the domain is all real numbers \((-abla, abla)\) and the range is also all real numbers \((-abla, abla)\). The inverse function \(f^{-1}(x) = \frac{x + 5}{4}\) has the same domain and range: domain \((-abla, abla)\) and range \((-abla, abla)\).

Key Concepts

One-to-One Functions

One-to-one functions are essential because they have a unique output for every input. This means that if you use two different inputs, you will always get two different outputs. Imagine a rule where each family member has their own unique coffee mug, and no one else uses it. That's a one-to-one relationship between people and mugs. For the function in the example, which is linear, having a non-zero slope suggests it is either always increasing or always decreasing, meeting our one-to-one criteria. This ensures the creation of an inverse function is possible, as it can be reversed to determine a single input for any given output.

Domain and Range

Understanding the domain and range is like understanding where a function starts and stops. The domain of a function defines all possible input values, while the range indicates all possible output values. Both the original function and its inverse, in our example, are linear equations.
This means:

• The domain of both functions is all real numbers \( (-\infty, \infty) \).
• The range for both functions is also all real numbers \( (-\infty, \infty) \).

Linear functions take any real number as an input and can produce any real number as an output, maintaining this consistency for their inverse as well.

Graphing Functions

Graphing functions help visualize their behavior, and it's especially neat with inverse functions. You start with the original function, like our example function \(y = 4x - 5\), and plot it on the coordinate plane. To find the inverse function's graph, you will reflect every point of the original over the line \(y = x\). It’s like looking in a mirror; the reflection should match the inverse function, which, in our example, is \(y = \frac{x + 5}{4}\).
This reflection property of inverses means that if you graph a function and its inverse on the same axes, they should be symmetrical concerning the line \(y = x\). This visual symmetry helps confirm you’ve computed the correct inverse function.

Linear Functions

Linear functions are characterized by their straight-line graphs, each defined by a basic form \(y = mx + b\), with \(m\) being the slope and \(b\) the y-intercept. The example function \(y = 4x - 5\) illustrates this, with a slope \(m = 4\) and y-intercept \(b = -5\).
Some important features of linear functions include:

• Constant slope, which tells you how steep the line is.
• Straight-line graph that makes them easy to predict and interpret.
• Always one-to-one when the slope is not zero, allowing them to have an inverse function.

If you understand these basics, graphing and finding inverses of linear functions becomes much simpler. They're the bread and butter of function types—reliable and straightforward to work with.