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Understanding the domain and range of a function is key to grasping how it behaves. The **domain** refers to all the possible input values (x-values) for which the function is defined. For different types of functions, this can vary:

• Exponential functions like \( f(x) = \left(\frac{1}{2}\right)^{x-1} - 1 \) have a domain of all real numbers, denoted as \( \mathbb{R} \). This is because exponential functions are defined for any real number input.
• Logarithmic functions such as \( g(x) = \ln |x| \) have a more restricted domain. Logarithms are only defined for positive inputs, so the domain is all non-zero values where \(|x| > 0\), which means \( x eq 0 \), represented as \((-\infty, 0) \cup (0, \infty)\).

The **range** is all the possible output values (y-values) a function can produce.

• For \( f(x) = \left(\frac{1}{2}\right)^{x-1} - 1 \), as \( x \to \infty \), \( f(x) \to -1 \). Hence, the range is \((-1, \infty)\).
• For \( g(x) = \ln |x| \), as \( |x| \to 0 \text{ from either direction, } g(x) \to -\infty \) and as \(|x| \to \infty, g(x) \to \infty\). So, the range is all real numbers: \( \mathbb{R} \).

Distinguishing domain and range helps in predicting where the graph will lie on the coordinate plane.

Exponential functions are defined by their base and exponent. A standard form is \( f(x) = a \cdot b^{x} \), where:

• "\(a\)" is the initial value or the vertical shift.
• "\(b\)" is the base, determining the growth or decay.

For example, in \( f(x) = \left(\frac{1}{2}\right)^{x-1} - 1 \):

• The base is \(\frac{1}{2}\), indicating the function is decreasing. The graph consistently moves downward because a smaller base (between 0 and 1) represents an exponential decay.
• The "-1" at the end shifts the entire graph down by 1 unit. The horizontal asymptote, a line the graph approaches but never reaches, moves from \( y = 0 \) to \( y = -1 \).

This shift also affects the *y-intercept*; usually found where \( x = 0 \), here the function has been shifted and crosses at \( f(1) = 0 \). Understanding these properties makes plotting exponential functions straightforward and helps in predicting their behavior over different x-values.

Logarithmic functions are the inverses of exponential functions, typically expressed as \( g(x) = \log_b(x) \). The natural logarithm, denoted as \( \ln x \), uses the base \( e \), an important mathematical constant approximately equal to 2.718.

• When dealing with \( g(x) = \ln |x| \), it's important to note the input, \(|x|\), which accounts for both positive and negative values of \( x \) (except zero).
• The logarithm graph is characterized by a vertical asymptote near zero. As \( x \) approaches zero, the output heads towards \( -\infty \).

A feature of logarithmic functions is their slow growth; even as \( \ln x \) increases, it does so more gradually compared to polynomial functions. With \( g(x) = \ln |x| \), outputs transition smoothly through the two branches. It captures changes in magnitude rather than the values themselves, which makes logarithms extremely useful in real-world applications, such as measuring pH, sound intensity, and in exponential decay processes. Understanding these characteristics affirms why logarithmic functions, like their function behavior, extend infinitely both negatively and positively.