Problem 80 - Calculus for Biology and Medicine

Question

Find the inverse of \(f(x)=\left(\frac{1}{2}\right)^{x}, x \geq 0\), together with its domain, and graph both functions in the same coordinate system.

Step-by-Step Solution

Answer

The inverse of \(f(x) = \left(\frac{1}{2}\right)^x\) is \(f^{-1}(x) = -\log_{2}(x)\), with a domain of \(x > 0\).

Step 1: Set up the equation

To find the inverse, we first write the given function equation as an equation with a variable. Since we are finding the inverse, set \( y = \left(\frac{1}{2}\right)^x \).

Step 2: Swap variables

Interchange \( x \) and \( y \) to set up the equation for the inverse. Now, our equation becomes \( x = \left(\frac{1}{2}\right)^y \).

Step 3: Solve for y

To solve for \( y \), convert the equation to logarithmic form. Using the properties of exponents and logarithms, we have \( y = \log_{\left(\frac{1}{2}\right)}(x) \).

Step 4: Simplify the logarithmic expression

Recall that \( \log_{\left(\frac{1}{2}\right)}(x) = -\log_{2}(x) \). Therefore, the inverse function is \( f^{-1}(x) = -\log_{2}(x) \).

Step 5: Determine the domain of the inverse

The domain of the inverse function \( f^{-1}(x) = -\log_{2}(x) \) is \( x > 0 \) as the logarithmic function is only defined for positive values.

Step 6: Determine the range of the original function

Since the domain of the inverse is the range of the original function, the range of \( f(x) = \left(\frac{1}{2}\right)^x \) is \( y > 0 \).

Step 7: Graph the functions

Plot both functions on a coordinate system. For \( f(x) = \left(\frac{1}{2}\right)^x \), the graph is a decreasing exponential curve. On the same graph, plot \( f^{-1}(x) = -\log_{2}(x) \), which is a decreasing logarithmic curve reflecting across the line \( y=x \).

Key Concepts

Understanding Exponential Functions

An exponential function is one where the variable is the power of a constant base. In the function \( f(x) = \left(\frac{1}{2}\right)^x \), the base here is \( \frac{1}{2} \). This base is a fraction, which means as \( x \) increases, \( f(x) \) will decrease, making it a decreasing exponential function. Exponential functions are fundamental in various fields such as biology, finance, and computer science since they model growth and decay processes.

• **Base greater than 1**: The function grows as \( x \) increases.
• **Base between 0 and 1**: The function decreases as \( x \) increases, like in our given function.

These functions have a few key characteristics:
- They never touch the x-axis; their value never actually becomes zero.
- For a base \( a \), positive \( x \) results in a range \( (0, \infty) \), and negative \( x \) results in \((0,\infty)\) when \( a < 1 \). This explains the range \( y > 0 \) for the given function.

Exploring Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It answers the question: to what power must a base be raised, to obtain a certain number? In our inverse function \( f^{-1}(x) = -\log_{2}(x) \), the base is 2 and we are seeking the exponent that gives us \( x \).
Here are some points to better understand logarithmic functions:

• The function \( y = \log_{b}(x) \) is only defined for \( x > 0 \).
• Logarithms convert multiplication into addition (e.g., \( \log(ab) = \log(a) + \log(b) \)).
• Negative logarithms, like \(-\log_{2}(x)\), reflect the curve, making it a decreasing function.

In the context of our problem, the inverse being a logarithmic function explains why the domain is strictly \( x > 0 \). Also, because it's the inverse, it will mirror the original exponential function about the line \( y = x \).

Domain and Range Essentials

The concepts of domain and range are crucial in understanding and graphing functions. The domain is the set of all possible input values \( x \) for which the function is defined, while the range is the set of all possible output values \( y \).
For our function \( f(x) = \left(\frac{1}{2}\right)^x \):

• **Domain**: \( x \geq 0 \). This is because powers of \( \frac{1}{2} \) remain defined only for non-negative values of \( x \).
• **Range**: \( y > 0 \). This reflects that no matter how large \( x \) gets, the function never reaches zero.

Similarly, for the inverse function \( f^{-1}(x) = -\log_{2}(x) \):

• **Domain**: \( x > 0 \). Logarithmic functions are undefined for non-positive numbers.
• **Range**: all real numbers \(( -\infty, \infty )\) because logarithms can produce any real number value.

Understanding this interaction between domain and range helps in plotting functions accurately and comprehending their limitations.