Problem 52 - Algebra 2

Question

Open-Ended Write a logarithmic function of the form \(y=\log _{b} x .\) Find its inverse function. Graph both functions on one set of axes.

Step-by-Step Solution

Answer

The inverse of the function \(y=\log _{2} x\) is the function \(y = 2^x\). When graphing these two functions, they will appear as reflections of each other over the line \(y = x\).

Step 1: Define the logarithmic function

Let's start by defining a simple logarithmic function of the form \(y=\log _{b} x\). For simplicity, let's consider \(b = 2\). Therefore, our function becomes \(y=\log _{2} x\)

Step 2: Find the inverse function

The inverse of a logarithmic function \(y=\log _{b} x\) is the exponential function \(x=b^y\). Substituting our base \(b=2\) into the equation, our inverse function becomes \(x=2^y\) or written in terms of \(x\), we can say \(y = 2^x\). This is the inverse of our original logarithmic function.

Step 3: Graph the functions

Both functions can now be graphed on the same axes to visually illustrate their relationship. When graphed, the function \(y=\log _{2} x\) should increase steadily, whereas its inverse \(y = 2^x\) should appear as a curve that increases sharply. Note that these two functions are reflections of each other in the line \(y = x\), which is a key property of inverse functions. The graphing can be easily done using a graphing calculator or some online plotting tool.

Key Concepts

Inverse Functions and Their Properties

Inverse functions are essential to understanding how one function undoes the action of another. If you have a function \(f(x)\), its inverse \(f^{-1}(x)\) will essentially reverse its effect. These functions are often pairs where the output of the initial function becomes the input of its inverse.Understanding inverses is crucial because they allow us to solve equations that involve functions. With logarithmic functions like \(y = \log_b x\), the inverse is an exponential function. In this case, the action of taking a logarithm is reversed by exponentiating.

• The inverse of \(y = \log_b x\) is \(y = b^x\).
• These functions "cancel" each other out, similar to how addition and subtraction cancel each other.

Additionally, a vital property of inverse functions is their reflection over the line \(y = x\). This means any point \((a, b)\) on the graph of a function will correspond to \((b, a)\) on its inverse, highlighting their symmetric relationship.

Understanding the Exponential Function

The exponential function is key when dealing with inverse functions of logarithms. It is defined as \(y = b^x\) where \(b\) is a positive real number, and never equal to one. This function grows rapidly and can model various real-world situations, such as population growth and radioactive decay.Some important aspects of exponential functions include:

• The base \(b\) determines the growth rate. Larger bases result in faster growth.
• An exponential function crosses the y-axis at the point \( (0, 1) \), since any base raised to the power of zero is one.
• Exponential functions exhibit continuous growth or decay depending on the base.

In our example, using a base of 2 for the logarithmic function means its inverse is \(y = 2^x\). This relates directly with our understanding of logarithms and how they operate as inverses to unravel exponentiation.

Graphing Logarithmic and Exponential Functions

Graphing logarithmic and exponential functions is an informative way to understand their properties. When you graph \(y = \log_2 x\), you’ll see a gradual increase, reflecting how slowly logarithmic functions grow.In contrast, when you graph its inverse, \(y = 2^x\), you notice a sharp increase characteristic of exponential growth.Here’s what you should consider:

• Both graphs are reflections over the line \(y = x\), confirming their inverse nature.
• The logarithmic function \(y = \log_2 x\) is only defined for \(x > 0\) and typically approaches negative infinity as \(x\) approaches zero from the positive side.
• The exponential graph \(y = 2^x\) extends infinitely in the positive y-direction, crossing the y-axis at \((0, 1)\).

Using a graphing calculator or online tool visualizes these functions clearly, making it easier to grasp how they interact as inverses. They beautifully demonstrate how one function expanding mirrors the compression effect of the other.