Problem 92 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The original function is \(f(x) = 2^{x}\) and the inverse function is \(f^{-1}(x) = \log_{2}(x)\). They can be graphed on the same coordinate system, with the latter being a reflection of the former over the line \(y = x\).

Step 1: Function Identification

The given function is \(f(x) = 2^{x}\) . Find the inverse function to this.

Step 2: Finding the Inverse Function

The inverse of a function can be found by switching the roles of \(y\) and \(x\). In this case, change \(f(x) = 2^{x}\) to \(x = 2^{y}\), and then solve for \(y\) . Use the property of logarithms to rewrite the equation as \(y = \log_{2}(x)\) . This equation is the inverse of \(f(x) = 2^{x}\)

Step 3: Graphing the Function and Its Inverse

Now that both functions are known, they can be graphed on the same coordinate plane. \(f(x) = 2^{x}\) is an exponential function that increases rapidly as x grows and approaches 0 as x decreases. The graph of \(y = \log_{2}(x)\) is a reflection of \(f(x) = 2^{x}\) over the line \(y = x\), since a function and its inverse are reflections of each other on the line \(y = x\). So, the graph of \(y = \log_{2}(x)\) increases at a slower rate as x grows and approaches negative infinity as x gets smaller.

Key Concepts

Exponential Functions

Exponential functions, like the one in the exercise, are mathematical expressions where the variable is an exponent. They typically take the form \( f(x) = a^x \), where \( a \) is a positive constant. In the original exercise, \( f(x) = 2^x \) represents an exponential function where the base is 2.

These functions are characterized by their rapid growth: as the value of \( x \) increases, \( f(x) \) rises steeply. Conversely, as \( x \) becomes more negative, \( f(x) \) approaches zero but never really reaches it, producing a horizontal asymptote at \( y=0 \).

Exponential functions are common in many real-world contexts, such as population growth, radioactive decay, and investment growth. Their increasing nature and the presence of an asymptote make them easy to spot when graphing.

• Key characteristic: Rapid increase or decrease.
• Intersects y-axis: Always at y = 1 unless there's a vertical shift.
• Horizontal Asymptote: Generally along the line y=0.

Logarithms

Logarithms are essentially the inverse operations to exponential functions. They help to solve equations involving exponents by converting them into a form where they can be more easily managed. In the solution, the function \( y = \log_2(x) \) is obtained by reversing the roles of \( x \) and \( y \) in \( f(x) = 2^x \).

The logarithm \( \log_2(x) \) answers the question: "To what power must 2 be raised, to equate to \( x \)?" Essentially, if \( 2^y = x \), then \( \log_2(x) = y \).

This inverse relationship with exponential functions means logarithmic functions grow much more slowly. This growth is essential in fields like computer science and information theory where multiplicative processes are analyzed.

• Basic Form: \( \log_b(x) \)
• Interpretation: What power does b need to become x?
• Non-Continuous at Origin: Undefined at \( x=0 \) and negative values.

Graphing Functions

Graphing functions is a visual method to understand mathematical relationships, allowing one to see trends, intersections, and growth patterns quickly. The exercise involves graphing \( f(x) = 2^x \) and its inverse \( y = \log_2(x) \), providing an excellent way to observe their inverse nature.

When graphing \( f(x) = 2^x \), expect a curve that starts near the horizontal asymptote \( y=0 \) for low \( x \) values, rising steeply as \( x \) increases.
The inverse function \( y = \log_2(x) \) will appear as a reflection of \( f(x) = 2^x \) along the line \( y = x \). It will increase slowly and is defined only for \( x > 0 \).

• Reflection Principle: Inverse functions are reflections over the line \( y=x \).
• Graphing Tool: Use tools like graph paper or software for perfect accuracy.
• Asymptotes: Indicative of function behavior as x approaches extreme values.