Problem 111 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Graph \(y=f(x), y=f^{-1}(x),\) and \(y=x\) (b) Determine the intervals where \(f\) and \(f^{-1}\) are increasing or decreasing. $$f(x)=\log _{3} x$$

Step-by-Step Solution

Answer

(a) Graphs: \( y = \log_{3} x \), \( y = 3^x \), \( y = x \). (b) Both \( f(x) \) and \( f^{-1}(x) \) are increasing on their respective domains.

Step 1: Understand the Graphs

The function given is \( f(x) = \log_{3} x \). The graph of \( y = f(x) \), which is \( y = \log_{3} x \), is an increasing logarithmic curve passing through the point (1,0) because \( \log_{3}1 = 0 \). The graph of \( y = x \) is a diagonal line that passes through the origin with a slope of 1.

Step 2: Finding the Inverse Function

The inverse function \( f^{-1}(x) \) can be found by swapping \( x \) and \( y \) in the equation and solving for \( y \). Start with \( x = \log_{3} y \). This can be rewritten as \( y = 3^x \). Therefore, \( f^{-1}(x) = 3^x \).

Step 3: Graph the Inverse Function

The graph of \( y = 3^x \), which represents \( y = f^{-1}(x) \), is an increasing exponential curve. It intersects the y-axis at (0,1) because \( 3^0 = 1 \). This curve is the reflection of \( y = \log_{3} x \) across the line \( y = x \).

Step 4: Determine Increasing/Decreasing Intervals for \( f(x) \)

The function \( f(x) = \log_{3} x \) is increasing for all \( x > 0 \). This is because the slope of the logarithmic function is positive for all \( x > 0 \).

Step 5: Determine Increasing/Decreasing Intervals for \( f^{-1}(x) \)

The function \( f^{-1}(x) = 3^x \) is also increasing for all \( x \). This is evident because an exponential function with a base greater than 1 (here, base 3) increases as \( x \) increases.

Key Concepts

Logarithmic Functions

Logarithmic functions are essential in mathematics, especially when dealing with operations that involve large ranges of numbers. A logarithm asks the question: "To what power must a specific base be raised to yield a particular number?" For example, in the context of the function \( f(x) = \log_{3} x \), it queries the power to which 3 must be raised to produce \( x \).
These functions typically display certain behaviors, such as passing through the point (1,0) on a graph, since any log with base \( b \), where \( b^1 = b \), results in zero. Their graphs are known for having a vertical asymptote along the y-axis (\( x = 0 \)), meaning they approach but never actually touch this line.

• Logarithmic functions are defined only for positive values of \( x \), as \( \,\log(b) \,\) of zero or a negative number is undefined.
• They extend infinitely to the right, increasing as \( x \) increases, and hence are classified as increasing functions.

Exponential Functions

Exponential functions are another fundamental type of mathematical function characterized by a constant base raised to a variable exponent. The function \( f^{-1}(x) = 3^x \) represents the exponential growth of numbers. Here, the base 3 leads to rapid growth as \( x \) increases.
The graph of an exponential function shows a distinct "rising" curve starting from the y-axis. For \( f^{-1}(x) = 3^x \), this graph will pass through point (0,1) since any number raised to zero is one. As \( x \) moves negatively, it approaches zero, showing a horizontal asymptote along the x-axis.

• Exponential functions are defined for all real numbers, creating a curve that is consistently increasing if the base is greater than one.
• They are rapid in growth, meaning values increase exponentially rather than linearly.

Graphing Functions

Graphing both logarithmic and exponential functions can help visualize their relationship and the concept of inverse functions. With the use of a coordinate plane, one can plot \( y = \log_{3} x \), its inverse \( y = 3^x \), and the line of reflection \( y = x \). This reflection confirms that when graphing inverse functions, the original function and its inverse are mirror images across the line \( y = x \).
Consider the following when graphing such functions:

• The function \( y = \log_{3} x \) will increase slowly, forming a smooth curve going upwards.
• Alternatively, \( y = 3^x \) rises sharply, depicting rapid exponential growth.
• The point where these graphs intersect involves determining when the function output equals the input.

This graphical representation aids in understanding the nature and behavior of these mathematical functions, underpinning their utility in solving real-world problems.