91Ó°ÊÓ

Understanding the domain of a function is critical because it tells us the set of all possible input values, often represented as x-values, for which the function is defined. For the function given by

\begin{flushleft}\[ y^{2} = \frac{1}{(x^{2} - 1)(x^{2} - 4)} \]\begin{flushleft},

we look for values of x that would make the denominator zero, as these are excluded from the domain. The zeros of the denominator occur at x = ±1 and x = ±2. Thus, the domain excludes these points and is represented as

\[ x \in (-\infty, -2) \cup (-2, -1) \cup (-1, 1) \cup (1, 2) \cup (2, \infty) \].

In simpler terms, the domain includes all real numbers except for -1, -2, 1, and 2. When plotting curves in calculus, identifying the domain is the first step as it gives structure to the graph and informs us where it exists on the x-axis.

A vertical asymptote is a vertical line x = a where the function tends towards positive or negative infinity as x approaches a. In our function,

\[ y^{2} = \frac{1}{(x^{2} - 1)(x^{2} - 4)} \],

the vertical asymptotes occur at the points where the denominator is zero, specifically at x = ±1 and x = ±2. This means the curve will get infinitely close to these lines but never actually touch them. Asymptotes provide crucial information about the behavior of a graph since they highlight regions where the function grows without bound. Not all functions have vertical asymptotes, but when they do, these are vital to include in the sketch of the graph.

Functions can exhibit different types of symmetry, which simplifies graphing the function. Symmetry in functions can be about the y-axis or the origin. If a function is symmetric about the y-axis, typically known as even symmetry, graphing one side provides a mirror image for the other side.

Our example function

\[ y^{2} = \frac{1}{(x^{2} - 1)(x^{2} - 4)} \]

is symmetric about the y-axis since replacing x with -x does not change the value of y. This symmetry is seen because each term with an x in the function is squared, making it even. When plotting, you only need to calculate points for either x ≥ 0 or x ≤ 0 and then reflect them across the y-axis to get the complete curve.

When it comes to curve sketching, a combination of all previously mentioned concepts comes into play. It's a multi-step process that generally involves the following:

• Identifying the domain of the function.
• Finding vertical and horizontal asymptotes, if they exist.
• Determining symmetry, to understand the behavior of the function and reduce the amount of computation needed.
• Calculating x-intercepts and y-intercepts.
• Analyzing the behavior of the function near the asymptotes and at infinity.
• Putting all these elements together to sketch the curve.

In our case, we've determined that the function given by

\[ y^{2} = \frac{1}{(x^{2} - 1)(x^{2} - 4)} \]

is substantially defined by its vertical asymptotes at x = ±1 and x = ±2 and its symmetry about the y-axis. There are no x-intercepts, and the y-intercepts are at (0, ±½). By putting together this information, we can sketch the curve by drawing a U-shaped graph in each interval between the asymptotes, making sure that the arms of the U's extend towards infinity as they approach the vertical asymptotes.