Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of \(f .\) Then use the graph of \(f\) to sketch the graph of \(g.\) $$\begin{aligned}&f(x)=15^{x}\\\&g(x)=\log _{15} x\end{aligned}$$

Step-by-Step Solution

Answer

The graph of function \(f(x)=15^x\) is a curve rising steeply from near the x-axis for negative \(x\), passing point (0, 1) and rising thereafter. The graph of function \(g(x)=\log_{15}(x)\) is a reflection of the graph of \(f(x)\) in line y=x and rises slowly for \(x > 0\).

Step 1: Sketch the graph of \(f(x)\)

Start by considering a few key points to plot for the function \(f(x) = 15^{x}\). Some useful points might be \((x, y) = (-1, 1/15), (0, 1), (1, 15)\). Remember that as \(x\) approaches negative infinity, \(15^{x}\) approaches 0, and as \(x\) approaches positive infinity, \(15^{x}\) grows very large. Draw a curve through these points, starting close to the x-axis for negative \(x\), passing through (0, 1) and rising steeply thereafter.

Step 2: Identify the properties of the function \(g(x) = \log_{15}x\)

\(\log_{a}b\) is the inverse function of \(a^{b}\). Therefore, the graph of \(g(x)\) will be the reflection of the graph of \(f(x)\) in the line \(y=x\). Also keep in mind that the function is not defined for \(x ≤ 0\).

Step 3: Sketch the graph of \(g(x)\)

Reflect the graph of \(f(x)\) in the line \(y=x\). This will give the graph of \(g(x)\). The reflection of (0, 1) is (1, 0), and the reflection of (1, 15) is (15, 1). For \(x > 0\), the graph will rise slowly. Connect these points with a smooth curve to complete the graph.

Key Concepts

Exponential Functions

Exponential functions are a class of mathematical functions often written in the form \(f(x) = a^x\), where \(a\) is a positive constant, and \(x\) is the variable. The base \(a\) is crucial because it determines how quickly the function grows as \(x\) increases.
For the exponential function \(f(x) = 15^x\), we observe the following important characteristics:

• The graph passes through the point (0, 1) since any number to the power of 0 is 1.
• As \(x\) increases, the function's value grows rapidly. This is the essence of exponential growth.
• Conversely, as \(x\) becomes negative, the function approaches 0 but never actually reaches it. This means the x-axis is a horizontal asymptote.

Visualizing the graph of an exponential function:
You can sketch the graph by plotting key points such as \((-1, \frac{1}{15})\), \((0, 1)\), and \((1, 15)\). With these points, you can draw a curve that rises quickly from left to right and flattens as it approaches the y-axis from the left.

The steep rise of the graph shows how exponential functions can quickly transform small changes in \(x\) into large changes in \(f(x)\).

Logarithmic Functions

Logarithmic functions provide a way to reverse exponential functions. For a function \(g(x) = \log_a x\), the variable \(x\) represents a value to which the base \(a\) must be raised to produce \(x\).
In the context of \(g(x) = \log_{15} x\):

• This function is defined only for positive values of \(x\) (i.e., \(x > 0\)).
• The point (1, 0) is included because \(15^0 = 1\).
• As \(x\) approaches infinity, \(g(x)\) increases but at a decreasing rate, depicting the curve’s slow ascent on the graph.
• The y-axis functions as a vertical asymptote, meaning the function approaches this line but never crosses it.

To sketch the graph of a logarithmic function:
The reflection of the exponential graph's points across the line \(y = x\) guides you. So, (0, 1) on the exponential graph reflects to (1, 0) on the logarithmic graph, and (1, 15) reflects to (15, 1).

The resulting graph rises gently for increasing \(x\) values, contrasting the sharp rise of exponential functions.

Inverse Functions

Inverse functions allow us to reverse one type of function to derive another. When we talk about exponential and logarithmic functions, understanding that they are inverse to each other is key. This means if the function \(f(x) = 15^x\) is given, then its inverse can be expressed as \(g(x) = \log_{15} x\).

• The graph of an inverse function is a reflection across the line \(y = x\). This line acts as a mirror, transforming every point on the graph of \(f(x)\) to its corresponding point on the graph of \(g(x)\).
• If a function passes through the point \((a, b)\), then its inverse will pass through the point \((b, a)\).

The concept of inverse functions is not just restricted to these two. Many functions have inverses, and finding them typically involves switching the roles of \(x\) and \(y\) in an equation and then solving for \(y\).

In practical terms, inverse functions help us solve problems where we need to "undo" a calculation or modeling process that was originally done using the original function.