Problem 68 - Algebra 2

Question

Find the inverse of each function. $$ y=\log (x+1) $$

Step-by-Step Solution

Answer

The inverse of the function \(y = \log(x + 1)\) is \(y = 10^x - 1\).

Step 1: Identify the Original Function

First, identify the original function, which in this case is \(y = \log(x + 1)\). Remember that the logarithm function base \(b\) is defined by: if \(y = \log_b(x)\) then \(b^y = x\)

Step 2: Swap \(x\)'s and \(y\)'s

To find the inverse of a function, swap the \(x\)'s and \(y\)'s. So, the function becomes \(x = \log(y + 1)\).

Step 3: Convert the Logarithmic equation into an Exponential equation

Now, convert the logarithmic equation into an exponential equation. Since we don’t have any base specified for the log, assume it to be 10 (this is often true for many scenarios). The equation becomes \(10^x = y+1\).

Step 4: Solve for \(y\)

Lastly, solve the equation for \(y\) to obtain the inverse function. In this case: \(y = 10^x - 1\). This is the inverse of the original function \(y = \log(x + 1)\).

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are mathematical expressions that help us understand how many times we need to multiply a certain number, called the base, to get another number. In simple terms, a logarithm indicates an exponent or power that a base number is raised to in order to reach a given value. For example, if you have the logarithmic function \( y = \log_b(x) \), it tells you that \( b^y = x \). Here, \( b \) is the base, \( y \) is the exponent, and \( x \) is the result when \( b \) is raised to the power of \( y \). This rearrangement is crucial for solving equations and finding inverse functions.

• The natural logarithm has a base of \( e \), where \( e \approx 2.718 \).
• The common logarithm uses base 10, often written simply as \( \log(x) \).

Logarithmic functions are very useful in situations involving exponential growth or decay. They can help solve real-world problems in science, finance, and data analysis.

Exploring Exponential Functions

Exponential functions are the "opposite" of logarithmic functions. They focus on the process of raising numbers to a power, rather than finding which power was used. The general form of an exponential function is \( y = b^x \), where \( b \) is the base and \( x \) is the exponent. As opposed to logarithmic functions that find the exponent, exponential functions find the resulting value when the base is raised to that power.

• An increase in the exponent leads to rapid growth in the function's value, making it useful for modeling populations or compound interest.
• When \( b \) is greater than 1, the function grows; when \( b \) is less than 1, it decays.

When discussing inverse functions, it's important to remember that the inverse of an exponential function is a logarithmic function. They "undo" each other, a concept used in the solution to the problem by converting a logarithm equation into an exponential one to solve for the inverse.

Understanding Function Transformation

Function transformation is an essential concept for understanding how functions can be manipulated. In the original problem, knowing how to transform functions allows us to find the inverse effectively.
The basic types of function transformations include:

• Translation: Shifting the graph horizontally or vertically. For example, \( y = \log(x + 1) \) is shifted one unit to the left compared to \( y = \log(x) \).
• Dilation: Stretching or compressing the graph.
• Reflection: Flipping the graph over an axis.

Function transformations allow us to see how alterations in the equation affect the graph and the suggested behavior. In the given exercise, recognizing initial forms and transformations is key. By understanding translations, we can reposition the function and find its inverse by switching \( x \) and \( y \), and then converting it.
Function transformation techniques are critical not just for finding inverses, but also for understanding the underlying patterns that various functions follow when adjusted.