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Rational functions are expressions that can be written as the ratio of two polynomials. These functions often have interesting characteristics, such as vertical asymptotes, holes, and horizontal asymptotes. Understanding the behavior of rational functions can help students predict and graph these functions accurately. For example, in the function \( g(x) = \frac{1}{x^2} \), the numerator is a constant 1, and the denominator is \( x^2 \). The properties of rational functions are crucial in calculus and pre-calculus courses, as they often feature in limit problems, integration, and differentiation.

An important concept in working with rational functions is identifying where the function is undefined. These undefined values occur when the denominator is zero because division by zero is not possible in mathematics. For \( g(x) = \frac{1}{x^2} \), the denominator becomes zero when \( x=0 \). Therefore, \( g(x) \) is undefined at \( x=0 \). To determine undefined values:

• Set the denominator equal to zero.
• Solve the equation to find the values of \( x \) that make the denominator zero.

This process helps identify potential vertical asymptotes. For \( g(x) = \frac{1}{x^2} \), solving \( x^2 = 0 \) reveals that \( x=0 \) is where the function is undefined.

Solving equations is the step-by-step process used to find the values of variables that make an equation true. In our context, we solve equations to find where the function's denominator is zero. Let's walk through the steps:
First, identify the denominator of the rational function. For \( g(x) = \frac{1}{x^2} \), it's \( x^2 \).
Next, set the denominator equal to zero: \( x^2 = 0 \).
Finally, solve for \( x \). For \( x^2 = 0 \), taking the square root of both sides gives \( x = 0 \).
This result tells us that the function \( g(x) \) becomes undefined at \( x=0 \). Identifying where the denominator is zero allows us to determine vertical asymptotes, which are places where the function 'spikes' to infinity or negative infinity on a graph.