Problem 34 - Precalculus with Limits

Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. \(f(x)=5^{x}, g(x)=\log _{5} x\)

Step-by-Step Solution

Answer

The exponential function \(f(x)\) will be an increasing curve across the points (-1, 1/5), (0, 1), and (1, 5) and the logarithmic function \(g(x)\) will be a curve increasing slowly across the points (1, 0), (5, 1). These graphs will reflect each other across the line \(y=x\), further confirming the fact that the logarithmic function and exponential function are inverses of each other.

Step 1: Plot the exponential function

The function \(f(x)=5^{x}\) is an exponential function. It is increasing and will cross the y-axis at \(y = 1\) (because any number raised to the power of 0 is 1). To plot the function, start by finding the y-coordinates when x equals, for example, -1, 0, and 1. Create a table for the function as follows: \n\n \[\mathrm{{x}}\,|\,-1\,,\,0\,,\,1 \]\n \[\mathrm{{f}}(\mathrm{{x}})\,|\,\frac{1}{5}\,,\,1\,,\,5\] \n\nPlot these points on the graph.

Step 2: Plot the logarithmic function

The function \(g(x)=\log _{5} x\) is a logarithmic function. It will cross the x-axis at \(x=1\) (because the logarithm of 1 to any base equals 0). The logarithm function approaches \(y = -\infty\) as \( x \to 0\) and it increases slowly. So, to plot the function, start by finding the x-coordinates when y equals, for instance, 0, and 1. You can create a table for the function as follows: \n\n \[\mathrm{{x}}\,|\,1\,,\,5\]\n \[\mathrm{{g}}(\mathrm{{x}})\,|\,0\,,\,1\] \n\nPlot these points on the graph.

Step 3: Sketch the function

You now have enough points to sketch the functions. For \(f(x)=5^{x}\), draw a curve that passes through the points (-1, 1/5), (0, 1), (1, 5) and continues towards \(\infty\) on the right and towards 0 on the left, above the x-axis. \n\nFor \(g(x)=\log _{5} x\), draw a curve that passes through the points (1, 0), (5, 1), and continues on upwards but at a decreasing rate. It should also approach the line \(y=-\infty\) as \(x\) decreases. The resulting sketch should clearly show that \(f(x)\) and \(g(x)\) are mirrored along the line \(y=x\).

Key Concepts

Exponential Functions

An exponential function is a type of mathematical expression where a constant base is raised to a variable exponent. Typically, these functions take the form \(f(x) = a^x\). It is crucial to understand how exponential functions behave:

• They increase or decrease rapidly, depending on the value of the base \(a\).
• If the base \(a\) is greater than 1, like 5 in the exercise, the function grows rapidly as \(x\) increases.
• Conversely, when 0 < \(a\) < 1, the function will decrease rapidly as \(x\) increases.
• The y-intercept is always at \(1\) because \(a^0 = 1\) for any \(a\) other than 0.

Let's apply this understanding to \(f(x)=5^{x}\). At \(x = 0\), \(f(x)\) equals 1. As we move in the positive direction on the x-axis, \(f(x)\) increases quickly. Moving towards negative \(x\), the function approaches 0 but never actually reaches it, thus staying above the x-axis.

Logarithmic Functions

Logarithmic functions serve as the inverse of exponential functions. The standard form of a logarithmic function is \(g(x) = \log_b x\), where \(b\) is the base, similar to the exponential function. Here's how logarithmic functions behave:

• Logarithmic functions grow slowly. For example, \(\log_{5} 1 = 0\) and \(\log_{5} 5 = 1\).
• The x-intercept is at 1, because the log of 1 to any base is 0.
• As \(x\) approaches infinity, \(g(x)\) increases slowly.
• As \(x\) approaches 0 from the right, the log function decreases indefinitely, tending towards negative infinity.

In the exercise, \(g(x) = \log_5 x\) reflects these behaviors. This function starts at the point (1, 0) because \(\log_5 1 = 0\). As \(x\) increases, \(g(x)\) grows at a decreasing rate.

Coordinate Plane

Plotting functions on the coordinate plane helps visually represent how each function behaves in relation to others. This representation aids in understanding their properties and relationships:

• Each point on the plane is defined by an \((x, y)\) coordinate pair.
• For exponential and logarithmic functions, we plot specific points calculated from the tables of values.
• These points guide the sketching of curves which represent the functions' general trends.
• Graphing both function types on the same plane reveals their inverse nature. The graph of \(f(x)\) and \(g(x)\) is mirrored across the line \(y = x\).

To draw the graph, we first mark the calculated points, such as (-1, 1/5), (0, 1), and (1, 5) for the exponential function. For the logarithmic function, we plot points like (1, 0) and (5, 1). Drawing smooth curves through these points provides us with accurate visual representations of both functions.