91Ó°ÊÓ

Understanding the domain of a function is crucial because it determines all the input values for which the function is defined. When dealing with rational functions, like \( g(t) = \frac{3t - 4}{t^2 - 9t + 8} \), we need to be extra attentive to the denominator. Rational functions can become undefined when their denominators equal zero, as division by zero is not allowed.

To find the domain of a rational function, follow these steps:

• Identify the denominator of the rational function.
• Set the denominator equal to zero and solve the resulting equation.
• Exclude the solutions from the set of all real numbers to find the domain.

For the given function, the domain is all real numbers except where \( t = 1 \) or \( t = 8 \), as these values cause the denominator to be zero. This leaves us with the domain in interval notation: \((\infty, 1) \cup (1, 8) \cup (8, \infty)\). Always remember that for rational functions, identifying problem values in the denominator helps you define where the function is valid.

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They are fundamental in algebra and arise naturally in various mathematical problems. In this exercise, the denominator of the rational function \( t^2 - 9t + 8 \) is a quadratic expression.

Quadratic equations can be solved using several methods:

• Factoring: If the quadratic can be factored into two binomial expressions, set each binomial to zero to find the solutions.
• Quadratic formula: For any quadratic equation \( ax^2 + bx + c = 0 \), the roots can be found using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Completing the square: This method involves rearranging the equation to form a perfect square trinomial, making it easier to solve.

In this exercise, the denominator was factored into \((t - 8)(t - 1) = 0\), providing solutions \( t = 8 \) and \( t = 1 \). Recognizing these roots allows us to determine where the original function is undefined.

Factoring is a technique used to simplify expressions and solve equations, especially useful for quadratic equations. The goal of factoring is to rewrite the expression as a product of simpler expressions.

For quadratic expressions like \( t^2 - 9t + 8 \), factoring involves finding two numbers that multiply to the constant term, 8, and add up to the linear coefficient, -9.

Steps for factoring a simple quadratic are:

• Write down the equation in standard form \( ax^2 + bx + c \).
• Determine pairs of numbers that multiply to \( c \) (the constant term).
• Out of these pairs, find the one that adds up to \( b \) (the coefficient of x).
• Rewrite the quadratic as a product of two binomials using these numbers.

In the original exercise, the quadratic \( t^2 - 9t + 8 \) was factored to \((t - 8)(t - 1)\), leading to the solutions \( t = 8 \) and \( t = 1 \). Mastery of factoring is key to cracking many algebraic problems and simplifying expressions effectively.