Problem 78 - Calculus for Biology and Medicine

Question

Find the inverse of \(f(x)=\left(\frac{1}{3}\right)^{x}, x \in \mathbf{R}\), together with its domain, and graph both functions in the same coordinate system.

Step-by-Step Solution

Answer

The inverse is \( f^{-1}(x) = \log_{\frac{1}{3}}(x) \), with a domain of \(x > 0\).

Step 1: Understand the Problem

We need to find the inverse of the function \(f(x)=\left(\frac{1}{3}\right)^{x}\). We also need to determine the domain of the inverse and graph both the function and its inverse.

Step 2: Find the Inverse Function

To find the inverse function, we need to solve the equation \(y = \left(\frac{1}{3}\right)^{x}\) for \(x\). Begin by taking the natural logarithm on both sides, resulting in \(\ln(y) = \ln\left(\left(\frac{1}{3}\right)^{x}\right)\). Use the power rule of logarithms to get \(\ln(y) = x \ln\left(\frac{1}{3}\right)\). Solving for \(x\), we find \(x = \frac{\ln(y)}{\ln\left(\frac{1}{3}\right)}\). Rewrite this as \(x = \log_{\frac{1}{3}}(y)\), so the inverse function \(f^{-1}(x)\) is \(\log_{\frac{1}{3}}(x)\).

Step 3: Determine the Domain of the Inverse

The domain of the inverse function \(f^{-1}(x) = \log_{\frac{1}{3}}(x)\) is determined by the logarithmic function \(\log_b(x)\), which is only defined for positive values of \(x\). Therefore, the domain of \(f^{-1}\) is \(x > 0\).

Step 4: Verify Results

Verify the results by composing \(f\) and its inverse \(f^{-1}\). We need to check that \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\). If \(f(x) = \left(\frac{1}{3}\right)^{x}\) and \(f^{-1}(x) = \log_{\frac{1}{3}}(x)\), then \(f(f^{-1}(x)) = \left(\frac{1}{3}\right)^{\log_{\frac{1}{3}}(x)} = x\) and \(f^{-1}(f(x)) = \log_{\frac{1}{3}}\left(\left(\frac{1}{3}\right)^x\right) = x\), confirms that the functions are indeed inverses.

Step 5: Graph Both Functions

To graph \(f(x) = \left(\frac{1}{3}\right)^{x}\), plot points for several values of \(x\) and sketch the curve, noticing its exponential decay to zero as \(x\) becomes larger. For \(f^{-1}(x) = \log_{\frac{1}{3}}(x)\), plot points for several positive values of \(x\) and sketch the curve, noting its general shape as it increases from negative infinity. Remember that the graph of the inverse is a reflection of the original function in the line \(y = x\). Display both graphs on the same coordinate system, and include the line \(y = x\) as a guide.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in which a constant base is raised to a variable exponent. The general form of an exponential function is \(f(x) = b^x\), where \(b\) is a positive real number not equal to 1. In our exercise, the function \(f(x) = \left(\frac{1}{3}\right)^x\) is an example of an exponential function. Here, the base is \(\frac{1}{3}\), indicating exponential decay, as \(x\) increases.These functions are characterized by their rapid growth or decay, depending on the base:

• If \(b > 1\), the function is increasing and represents exponential growth.
• If \(0 < b < 1\), the function is decreasing and represents exponential decay.

Understanding the behavior of exponential functions is crucial when graphing and analyzing their properties.Exponential functions exhibit unique features:

• The graph approaches zero but never actually reaches it, known as an asymptote.
• The y-intercept of \(f(x) = b^x\) is always 1 because \(b^0 = 1\).

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and they allow us to solve for the exponent in expressions of the form \(b^x = y\). Mathematically, the logarithmic function is denoted as \(log_b(y) = x\), where \(b\) is the base.In the case of our inverse function \(f^{-1}(x) = log_{\frac{1}{3}}(x)\), the logarithmic base is \(\frac{1}{3}\). This means that we're finding the exponent that the base \(\frac{1}{3}\) needs to be raised to in order to yield \(x\).Important characteristics of logarithmic functions include:

• They are only defined for positive values of \(x\), due to the nature of logarithms.
• The graph of a logarithmic function passes through the point (1,0) because \(log_b(1) = 0\) for any base \(b\).
• They increase without bound as \(x\) increases.

Recognizing the relationship between exponential and logarithmic functions helps in solving equations and graphing them effectively.

Domain and Range

In mathematics, the domain and range of a function describe the set of possible input values (domain) and output values (range) respectively.The domain of the exponential function \(f(x) = \left(\frac{1}{3}\right)^x\) includes all real numbers. This is because you can substitute any real number for \(x\) and the function remains valid.For its inverse, \(f^{-1}(x) = \log_{\frac{1}{3}}(x)\), the domain is more limited:

• Since logarithmic functions are only defined for positive values of \(x\), the domain of \(f^{-1}(x)\) is \((0, \infty)\).

Ranges of these functions are:

• For the exponential function \(f(x)\), the range is \((0, \infty)\), as it only outputs positive values.
• The range of the inverse logarithmic function \(f^{-1}(x)\) is all real numbers \((-\infty, \infty)\), because a logarithm can take any real value.

Domain and range are essential for understanding how functions behave and what kind of transformations can be applied.

Graphing Functions

Graphing helps visualize the relationship between the variables in a function. When graphing both an exponential function and its inverse logarithmic function, some interesting properties become clear.For an exponential decay function like \(f(x) = \left(\frac{1}{3}\right)^x\):

• The graph will show a rapid decrease as \(x\) increases, approaching zero but never reaching it.
• The y-intercept occurs at (0, 1).

The inverse function \(f^{-1}(x) = \log_{\frac{1}{3}}(x)\) will be a reflection of \(f(x)\) across the line \(y = x\):

• The graph begins at the point (1, 0) and increases without bound as \(x\) becomes large.
• Notice that as \(x\) approaches zero from the right, the logarithmic function decreases toward negative infinity.

Plotting these graphs on the same coordinate system is helpful to see the inverse relationship. They show a mirror image along the line \(y=x\), highlighting how each point on one graph has a corresponding point on the other graph in the opposite direction.