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Understanding the domain of a function is crucial in calculus and function analysis. The domain of a function refers to all the input values (usually x-values) for which the function is defined. A function like \( f(x) = \ln(x^2) \) requires its argument to be strictly positive—meaning we need to determine when \( x^2 > 0 \). This condition is satisfied for all real numbers except zero, thus the domain is \( (-\infty, 0) \cup (0, \infty) \).

In contrast, the function \( g(x) = e^{-x^2} \) has no such restrictions since exponential functions are well-defined for all real numbers. Thus, its domain is all real numbers, \((-\infty, \infty)\). Understanding these domains helps in predicting and explaining the behavior of the function in various intervals:

• Check for restrictions like square roots, logarithms, and denominators in fractions.
• Remember exponential functions are defined everywhere.

The range of a function refers to all the potential output values the function can produce, typically involving y-values. Calculating the range helps us understand where the function can "take us."

For \( f(x) = \ln(x^2) \), consider that \( x^2 \) takes on all positive values as \( x \) traverses its domain. Since the range of \( \ln(u) \) when \( u > 0 \) is the entire set of real numbers, the range of \( f(x) \) will be \( (-\infty, \infty) \). This indicates the function can produce any real value as output.

In the case of \( g(x) = e^{-x^2} \), the expression \( -x^2 \) results in a non-positive exponent. The smallest value, \( e^0 = 1 \), occurs at \( x = 0 \). As \( x \) deviates from zero, \( -x^2 \) decreases, causing \( e^{-x^2} \) to approach zero but never actually reaching it. Thus the range for \( g(x) \) becomes \( (0, 1] \). In finding the range:

• Evaluate limits for boundary behavior.
• Consider the inherent properties of the function type.

Drawing and analyzing the graph of a function is a visual way to understand its behavior and characteristics. When graphing \( f(x) = \ln(x^2) \), remember that it mirrors a standard logarithmic curve but is symmetric about the y-axis because \( \ln(x^2) = 2\ln(|x|) \). It diverges to \(-\infty\) as \( x \to 0 \) from either side and ascends to infinity as \( |x| \) increases.

The graph for \( g(x) = e^{-x^2} \) is a classic bell curve or Gaussian shape, peaking at 1 when \( x = 0 \) and tapering off towards zero as \( |x| \) becomes larger. This dampened curve is used extensively in statistics for normal distribution models.

To successfully graph functions:

• Identify symmetry properties to simplify plotting.
• Look for asymptotic behavior, like approaching a line without crossing.
• Find critical points where the function peaks or troughs.