91Ó°ÊÓ

A rational function is any function that can be expressed as the ratio of two polynomial functions. For example, a general form of a rational function is: \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are special because they can have asymptotes: lines that the graph of the function approaches but never actually touches. Understanding the behavior of the numerator and the denominator is crucial when working with rational functions. Examples:

• \( \frac{x^2 + 2x + 1}{x - 3} \)
• \( \frac{2x^3 - x^2 + 5}{4x - 7} \)

Function evaluation simply involves finding the output of a function for a particular input value. Given \( h(x) = \frac{x+1}{3x-1} \), to evaluate this function at say \( x = 2 \), you substitute \( x=2 \) into the function: \( h(2) = \frac{2+1}{3\cdot2-1} = \frac{3}{5} \). This process also applies when the input is another function or expression. In the exercise, we substitute \( \frac{1}{x^2} \) into \( h(x) \): \( h\left( \frac{1}{x^2}\right) = \frac{\frac{1}{x^2} + 1}{3 \cdot \frac{1}{x^2} - 1} \). This substitution results in a new rational function.

Algebraic simplification makes complex expressions more manageable. Let's break down the steps using the exercise. We start with: \[ h\left( \frac{1}{x^2} \right) = \frac{\frac{1}{x^2} + 1}{3 \cdot \frac{1}{x^2} - 1} \] First, simplify the numerator: \[ \frac{1}{x^2} + 1 = \frac{1 + x^2}{x^2} \] Next, simplify the denominator: \[ 3 \cdot \frac{1}{x^2} - 1 = \frac{3}{x^2} - 1 \] Now, our function looks like: \[ \frac{\frac{1 + x^2}{x^2}}{\frac{3}{x^2} - 1} \] Simplify this by multiplying the numerator and denominator by \( x^2 \): \[ \frac{1 + x^2}{3 - x^2} \] This is much simpler and easier to work with. Mastering algebraic simplification is crucial for solving complex rational functions.