Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane.\(f(x)=3^{x}, g(x)=\log _{3} x\)

Step-by-Step Solution

Answer

The graphs of the exponential function \(f(x)=3^x\) and the logarithmic function \(g(x)=\log_3 x\) intersect at the point (1,1). The graph of \(f(x)\) starts at (0,1) and increases rapidly for \(x>0\). The graph of \(g(x)\) starts from very negative values as x approaches 0 from the right, crosses the x-axis at (1,0), and increases slowly as x grows beyond 1. Their graphs are mirror images of each other about the line \(y=x\).

Step 1: Sketch Exponential Function

To sketch the exponential function \(f(x) = 3^x\), begin at point (0, 1) because any number to the power of 0 is 1. This function is an increasing function because the base (3) is greater than 1. So for \(x < 0\), the function \(y = 3^x\) approaches 0, and for \(x > 0\), the function increases rapidly.

Step 2: Sketch Logarithmic Function

Then, sketch the logarithmic function \(g(x) = \log_3 x\). Logarithmic function has a horizontal asymptote at \(x=0\). This function will start from very negative values at x approaches 0 (from the right), cross the x-axis at (1,0) as \(\log _{3} 1 = 0\), and increase slowly for \(x > 1\). Note that logarithm is only defined for \(x > 0\).

Step 3: Identify the Intersection Point

The intersection point of these two graphs is (1,1) because \(\log_3 (3^1) = 1\). That is, \(g(f(1)) = 1\).

Step 4: Merge the Two Graphs

Now consolidate these two sketches into a single graph on the same coordinate plane. Annotation of key points and drawing arrows at the end of the curves to indicate the direction in which the function extends beyond the graph, completes the sketch.

Key Concepts

Exponential Function

An exponential function is characterized by its rapid growth. The general form is given by \( f(x) = a^x \), where \( a \) is the base and is a positive number not equal to 1. In our exercise, \( f(x) = 3^x \) is an example. This function starts at a point \((0, 1)\) on the y-axis because \( 3^0 = 1 \). From there, it rises sharply because the base, 3, is greater than 1.
Understanding the behavior of exponential functions is crucial:

• For \(x < 0\), \(f(x)\) approaches zero. The curve gets closer and closer to the x-axis but never touches it.
• For \(x > 0\), \(f(x)\) increases rapidly. The larger the x, the higher the value of \(f(x)\). This characteristic creates a steep curve.
• Exponential functions are continuous and smooth. There are no sharp turns or holes in their graphs.

This function's rapid growth makes it useful in many real-world scenarios, such as calculating compound interest, population growth, and more. Observing a graph will show a curve that bends sharply upwards when moving from left to right on the coordinate plane.

Logarithmic Function

Unlike exponential functions, logarithmic functions grow slowly. The logarithmic function \( g(x) = \log_3 x \) is the inverse of the exponential function \( f(x) = 3^x \). This means it flips the input-output relationship of \( f(x) \).
Some important features of logarithmic functions include:

• The function is only defined for positive values of x, \( x > 0 \).
• It crosses the x-axis at \( (1, 0) \) because \( \log_3 1 = 0 \).
• As \( x \) approaches zero from the positive side, \( g(x) \) drops down to negative infinity, creating a vertical asymptote at \( x = 0 \).
• The function increases, but at a decreasing rate, as \( x \) increases beyond 1.

Logarithmic functions are essential in situations where we need to manage exponential growth inversely, such as calculating time in decay processes or understanding the Richter scale for earthquakes. On a graph, they will appear to curve upward gently rather than abruptly.

Graph Sketching

Graph sketching brings together all the concepts of functions visually. When you sketch both an exponential function like \( f(x) = 3^x \) and a logarithmic function like \( g(x) = \log_3 x \) on the same coordinate plane, you see how they relate.
Points to consider when sketching graphs:

• Identify significant points, such as (0,1) for exponential functions and (1,0) for logarithmic functions. These are key because they often form the baseline of your graph.
• Highlight the asymptotes, such as the x-axis for the exponential and \( x = 0 \) for the logarithmic function. Asymptotes are lines that the graph approaches but never touches.
• Note the intersection point, (1,1) in our exercise. This is a crucial point because both functions share it.
• Extend the graphs with arrows to show current trends in growth or decline beyond the drawn portion.

When combined, these graphs not only demonstrate symmetry due to their inverse nature but also provide a comprehensive view of how different mathematical concepts interlock through visual representation. This understanding aids in interpreting behaviors of complex systems in natural and scientific fields.