Problem 63 - College Algebra

Question

The exponential function \(y=2^{x}\) is one-to-one and has an inverse function. Try finding the inverse function by exchanging \(x\) and \(y\) and solving for \(y .\) Describe the difficulty that you encounter in this process. What is needed to overcome this problem?

Step-by-Step Solution

Answer

The inverse function of \(y = 2^{x}\) is \(y = \log_{2}{x}\)

Step 1: Swap \(x\) and \(y\)

First, we will swap \(x\) and \(y\) in the equation \(y = 2^{x}\). This gives us \(x = 2^{y}\). The reason we swap \(x\) and \(y\) is because in an inverse function, the roles of \(x\) and \(y\) are reversed.

Step 2: Solve for \(y\)

Next, we try to solve for \(y\) from the equation \(x = 2^{y}\). However, in this situation, unlike functions where \(y\) can be solved directly from the equation, figuring out how to solve for \(y\) when it's in the exponent can be challenging. We need to use a mathematical tool known as logarithm. A logarithm allows us to bring down the exponent and thus make \(y\) solvable from the equation.

Step 3: Apply Logarithm

We will use the logarithm to base 2, as our equation is in base 2. So, we get: \(\log_{2}{x} = y\). And so, the inverse function for \(y = 2^{x}\) is \(\log_{2}{x}\).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a variable appears in the exponent. This type of function is of the form \(y = b^x\), where \(b\) is a positive constant called the base, and \(x\) is the exponent. In our original exercise, the function is \(y = 2^x\). This indicates that for every increase in \(x\), \(y\) grows by a factor of 2.

Exponential functions are known for their rapid growth. They are used to describe phenomena such as population growth, radioactive decay, and interest calculations in finance. One of the key features of exponential functions is their "one-to-one" nature, which means that for every \(x\), there is one unique \(y\), and vice versa. This characteristic is important, as it allows us to find an inverse function.

• Grows rapidly: An exponential function increases very quickly.
• Base Denotes Growth Rate: In \(b^x\), \(b > 1\) implies growth, while \(0 < b < 1\) means decay.

Logarithm

Logarithms are the inverse operations of exponential functions. They help us solve equations where the variable is an exponent. In our scenario, when we swap \(x\) and \(y\) to find the inverse of \(y = 2^x\), we face the challenge of extracting \(y\) from the equation \(x = 2^y\).

The logarithm provides a solution: it allows us to move the exponent down and solve for \(y\). Essentially, if \(b^y = x\), then \(y = \log_b(x)\). Applying this to our equation, \(x = 2^y\), we use logarithm base 2 to rewrite it as \(y = \log_2(x)\).

Logarithms have several important properties:

• Product Rule: \(\log_b{(mn)} = \log_b{m} + \log_b{n}\)
• Quotient Rule: \(\log_b{\left(\frac{m}{n}\right)} = \log_b{m} - \log_b{n}\)
• Power Rule: \(\log_b{(m^n)} = n \cdot \log_b{m}\)

These properties make logarithms powerful tools for simplifying complex equations.

One-to-One Function

A one-to-one function is a type of function where each output value (\(y\)) is associated with exactly one input value (\(x\)), and each input value maps to a single output. This unique matching characteristic is crucial for a function to have an inverse function.

In the context of our original problem, the function \(y = 2^x\) is one-to-one because no two different \(x\) values will produce the same \(y\). This means that we can reverse the function and find an inverse function. If a function is not one-to-one, it cannot have an inverse, or its inverse will not be a function.

To determine if a function is one-to-one, you can use the Horizontal Line Test. If any horizontal line intersects the graph of the function at most once, the function is one-to-one. This test confirms that exponential functions, like \(y = 2^x\), are indeed one-to-one.

• Unique Mapping: Every \(x\) has a unique \(y\) and vice versa.
• Inverse Exists: Only one-to-one functions can have inverse functions.

Understanding if a function is one-to-one is fundamental before attempting to find its inverse.