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The domain of a rational function comprises all real numbers, except for those that make the denominator zero. This is because division by zero is undefined in mathematics. To find these exclusions, we set the denominator equal to zero and solve for any values of \(x\) that satisfy this equation.

In our example, the denominator is \(4x^2 - 1\). By setting this to zero and solving,

• \(4x^2 = 1\)
• \(x^2 = \frac{1}{4}\)
• \(x = \pm \frac{1}{2}\)

Thus, the domain is all real numbers except \(x = \pm \frac{1}{2}\). It's crucial to identify the domain so you know where the function is defined and valid.

Intercepts give us critical points where the graph of the function intersects the axes.

**Finding \(x\)-Intercepts:**
For \(x\)-intercepts, the function \(y\) equals zero. Set the numerator \(x^2 - 9\) equal to zero and solve. This gives:

• \(x^2 = 9\)
• \(x = \pm 3\)

Thus, the \(x\)-intercepts are at points \((3, 0)\) and \((-3, 0)\).

**Finding \(y\)-Intercept:**
To find the \(y\)-intercept, substitute \(x = 0\) into the function:

• \(y = \frac{0^2 - 9}{4\cdot0^2 - 1} = \frac{-9}{-1} = 9\)

The \(y\)-intercept is \((0, 9)\). These points are essential for sketching the graph accurately.

Asymptotes are lines that a graph approaches but never touches. There are two types relevant here: vertical and horizontal.

**Vertical Asymptotes:**
These occur at values of \(x\) that make the denominator zero, without making the numerator zero. Solving \(4x^2 - 1 = 0\) gives:

• \(x = \pm \frac{1}{2}\)

These are the vertical asymptotes: \(x = -\frac{1}{2}\) and \(x = \frac{1}{2}\). Graphically, they segment the graph.

**Horizontal Asymptotes:**
These depend on the degree of the numerator and denominator. If both have the same degree, the horizontal asymptote is the ratio of their leading coefficients. Here, the degrees are both 2, so

• \(y = \frac{1}{4}\)

This denotes the line the graph settles into as \(x\) approaches infinity.

A graphing utility is an invaluable tool for visualizing and confirming your analytical work. It allows you to plot complex functions and quickly verify intercepts and asymptotes.

When using a graphing utility, plot your function: \(y = \frac{x^2 - 9}{4x^2 - 1}\). Notice how the graph behaves near the intercepts and asymptotes.

• Check if vertical asymptotes appear at \(x = \pm \frac{1}{2}\).
• Confirm a horizontal asymptote at \(y = \frac{1}{4}\).
• Ensure the intercepts \((3,0), (-3,0)\) and \((0,9)\) are displayed.

By comparing these visual results with the analytical findings, you can strengthen your understanding and interpretation of rational functions.