Problem 79 - Calculus for Biology and Medicine

Question

Find the inverse of \(f(x)=2^{x}, x \geq 0\), together with its domain, and graph both functions in the same coordinate system.

Step-by-Step Solution

Answer

The inverse of \(f(x)=2^x\) is \(f^{-1}(x)=\log_2(x)\), with domain \(x > 0\).

Step 1: Recognize Function and Domain

We are given the function \(f(x) = 2^x\) and the domain is \(x \geq 0\). This is an exponential function where \(x\) is the exponent.

Step 2: Set Function Equal to y

To find the inverse, we first set \(f(x)\) equal to \(y\), so we have \(y = 2^x\).

Step 3: Swap Variables

The next step to finding the inverse is to swap \(x\) and \(y\). So we change the equation to \(x = 2^y\).

Step 4: Solve for y

To solve \(x = 2^y\) for \(y\), take the logarithm base 2 of both sides. This gives us \(y = \log_2(x)\).

Step 5: Identify the Inverse Function

Thus, the inverse of \(f(x) = 2^x\) is \(f^{-1}(x) = \log_2(x)\).

Step 6: Determine Domain of Inverse Function

Since the original function has a domain of \(x \geq 0\), the range for this is \(y \geq 1\). Therefore, the domain of \(f^{-1}(x)\) is \(x > 0\).

Step 7: Graphing

Graph both \(f(x) = 2^x\) and \(f^{-1}(x) = \log_2(x)\) on the same set of axes. The graph \(f(x) = 2^x\) is an exponential curve starting from the point (0,1) and moving upwards. The inverse \(f^{-1}(x) = \log_2(x)\) is the reflection of this curve across the line \(y = x\).

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics, characterized by constant bases raised to variable exponents. In our exercise, the function given is an exponential function described as \(f(x) = 2^x\). Here, the base, 2, is consistent, while the exponent, \(x\), changes.
Key features of exponential functions include:

• They grow rapidly as the value of \(x\) increases due to repeated multiplication.
• Their graphs are always increasing for positive bases greater than 1, unless transformations are applied.
• They are distinguished by smooth, continuous curves.

Understanding the exponential function involves recognizing its rapid growth and its unique curve that starts from a certain point (in this case (0, 1)) and extends to infinity. The exponential growth property is pivotal for applications ranging from population dynamics to finance.
In mathematical terms, transforming and manipulating exponential functions involves operations such as finding inverses, where the concept of logarithms becomes especially useful.

Graphing Functions

Graphing functions allows us to visualize how they behave and interact across different values of \(x\). For the function \(f(x) = 2^x\), the graph begins at the point (0, 1) and moves upward steeply, illustrating its exponential nature.
When graphing:

• The y-intercept of the exponential graph is always at (0,1) when the base is greater than 1.
• The curve is smooth and gradually steepens.
• For exponential growth, the graph does not touch the x-axis; it approaches but never reaches zero.

Graphing both an exponential function and its inverse on the same set of axes helps to see the symmetry. The inverse function, in this case \(f^{-1}(x) = \log_2(x)\), appears as a reflection of \(f(x)\) across the line \(y = x\). This reflection property is a hallmark of inverse functions, highlighting how they "undo" the effects of the original.

Logarithmic Functions

Logarithmic functions serve as the inverse operations of exponential functions. The inverse of \(f(x) = 2^x\) is \(f^{-1}(x) = \log_2(x)\), showing how logarithms "reverse" the operation of exponentiation.
Here are some important insights about logarithmic functions:

• The domain of a logarithmic function like \(f^{-1}(x) = \log_2(x)\) is \(x > 0\), as you cannot take the logarithm of zero or a negative number.
• The graph of a logarithmic function is a curve that increases slowly and passes through (1, 0), where the logarithm of 1 is always zero, given any base \(b > 1\).
• It approaches the y-axis from the right but never touches it, reflecting its undefined nature at \(x = 0\).

Understanding logarithmic functions requires recognizing their purpose as solutions to the question: "To what power must the base be raised, to obtain this number?" This question is foundational in solving many mathematical problems, especially in contexts where rates and scales are involved.