91Ó°ÊÓ

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It is often written in the form \( y = a^x \), where \( a \) is a positive constant called the base, and \( x \) is the exponent. In general, exponential functions depict rapid growth or decay.

The function \( f(x) = \left( \frac{1}{2} \right)^{x-1} - 1 \) is a specific case of an exponential function. Here, the base is \( \frac{1}{2} \), indicating that the function models exponential decay rather than growth, since the base is less than 1.

Key characteristics of exponential functions include:

• Domain: Usually, the domain of an exponential function is all real numbers \( \mathbb{R} \).
• Range: Depending on whether the function is shifted vertically, the range can vary. For example, for this function, the range after a downward shift is \((-1, 0]\).
• Graph Behavior: When graphed, exponential decay will approach zero but never actually reach it, appearing as a curve that gets closer and closer to the x-axis on the positive side and to \( y = -1 \) on the negative side.

The natural logarithm function, denoted as \( \ln(x) \), stands out because it uses the mathematical constant \( e \) (approximately 2.718) as its base. This function is essential in calculus, commonly representing the time needed to reach a certain level of growth.

For the function \( g(x) = \ln |x| \), the use of \( |x| \) means the logarithm takes the absolute value of \( x \), making the function defined for all non-zero values of \( x \), both positive and negative.

Essential traits of the natural logarithm include:

• Domain: The domain of \( \ln |x| \) is \( x \in (-\infty, 0) \cup (0, \infty) \), capturing all real numbers except zero.
• Range: As with all logarithms, \( \ln |x| \) can take any real value. Therefore, its range is all real numbers \( \mathbb{R} \).
• Graph Behavior: The graph for \( g(x) = \ln |x| \) closely resembles the graph of \( \ln(x) \) for positive values of \( x \) but is mirrored across the y-axis for negative values of \( x \). The function tends toward negative infinity as \( x \) nears zero from either side.

Graphing a function provides a visual representation of its behavior over its domain. When you sketch a graph, you illustrate how the function values change with the input values. For example, understanding the graph's trend helps identify limits and asymptotic behavior.

When sketching the graph of \( f(x) = \left( \frac{1}{2} \right)^{x-1} - 1 \), you can expect to see a curve that dips as a declining exponential form. Approach the horizontal asymptote \( y = -1 \) from above as \( x \to -\infty \), showing the nature of decay.

On the other hand, the graph of \( g(x) = \ln |x| \) is unique as:

• It behaves similar to \( \ln(x) \) on the right of the y-axis, increasing slowly as \( x \) grows.
• It is reflected over the y-axis to account for negative \( x \)-values, demonstrating symmetry.
• The curve drops sharply to \( -\infty \) as \( x \) approaches zero from either direction, reflecting the function's undefined nature at \( x = 0 \).

Graphing these functions without technology encourages a deeper understanding of their properties and enhances skill in visualizing mathematical relationships.