Problem 7 - Algebra 2 and Trigonometry

Question

\(\ln 3-10 :\) a. For each \(f(x),\) write an equation for \(f^{-1}(x),\) the inverse function. b. Sketch the graph of \(f(x)\) and of \(f^{-1}(x) .\) $$ f(x)=\left(\frac{1}{2}\right)^{x} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \frac{\ln x}{\ln \left(\frac{1}{2}\right)}\). Graphs not provided.

Step 1: Set up the inverse

Let \( y = \left(\frac{1}{2}\right)^x \). To find the inverse, swap \( x \) and \( y \): \( x = \left(\frac{1}{2}\right)^y \).

Step 2: Solve for y

Take the natural log: \( \ln x = y \ln\frac{1}{2} = -y\ln 2 \).
Therefore \( y = -\frac{\ln x}{\ln 2} = \log_{1/2} x \).

Step 3: State the result

\( f^{-1}(x) = \frac{\ln x}{\ln(1/2)} = -\frac{\ln x}{\ln 2} = \log_{1/2} x \).
The graph of \( f \) is an exponential decay curve; \( f^{-1} \) is its reflection over \( y = x \).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function in which a constant base is raised to a variable exponent. A key feature of these functions is their rapid growth or decay. An exponential function takes the form \( f(x) = a^x \), where \( a \) is a positive constant and \( x \) is the exponent. In our exercise, the function is \( f(x) = \left(\frac{1}{2}\right)^x \), which represents exponential decay because \( \frac{1}{2} \) is a fraction less than one. Therefore, as \( x \) increases, \( f(x) \) decreases rapidly. Important properties of exponential functions include:

• The base \( a \) must be positive and not equal to 1.
• If \( a > 1 \), the function represents exponential growth.
• If \( 0 < a < 1 \), the function represents decay, as in our example.
• Exponential functions have a horizontal asymptote, typically the x-axis, where \( f(x) \) approaches 0 as \( x \) approaches infinity.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. Understanding them is crucial for solving equations involving exponential functions, especially when finding inverse functions. The general form of a logarithmic function is \( y = \log_a(x) \), meaning \( a^y = x \). Logarithms tell us the power to which a certain base must be raised to yield a given number.In our solved problem, we use the natural logarithm \( \ln \), which is a logarithm with the base \( e \), a mathematical constant approximately equal to 2.718. To find the inverse of our exponential function \( f(x) = \left(\frac{1}{2}\right)^x \), we first express it in the form of \( y = \left(\frac{1}{2}\right)^x \). By switching \( x \) and \( y \), and using logarithms, we determine \( f^{-1}(x) = \frac{\ln x}{\ln \left(\frac{1}{2}\right)} \).Some key characteristics of logarithmic functions are:

• They reverse the operations of exponential functions.
• The function is defined only for \( x > 0 \).
• Their graphs are a reflection over the line \( y = x \) of their corresponding exponential functions.

Graph Sketching

Graph sketching of functions and their inverses provides insight into the nature and behavior of these functions. For the functions \( f(x) = \left(\frac{1}{2}\right)^x \) and its inverse \( f^{-1}(x) = \frac{\ln x}{\ln \left(\frac{1}{2}\right)} \), let's consider their graphing.The graph of the exponential function \( f(x) \) will generally:

• Start above the x-axis and move towards it as \( x \) increases (exponential decay).
• Have a horizontal asymptote along the x-axis, which it approaches but never touches.
• Pass through the point (0, 1) because any non-zero number raised to the power of zero is one.

For the inverse function graph \( f^{-1}(x) \):

• The graph will be a reflection of \( f(x) \) across the line \( y = x \).
• Since it is a logarithmic function, it rises slowly and has its own characteristic curve.
• The graph passes through the point (1, 0), consistent with the property of logarithms where \( \log_a(1) = 0 \) for any base \( a \).

Graphing these functions helps visualize their inverse relationship and is a useful tool for understanding exponential and logarithmic behavior.