Problem 8 - College Algebra

Question

Find the domain of each function. $$ g(x)-\frac{2}{x^{2}+x-12} $$

Step-by-Step Solution

Answer

The domain of the function \(g(x) = \frac{2}{x^{2}+x-12}\) is all real numbers except x = 3 and x = -4.

Step 1: Define the function

The given function is \(g(x) = \frac{2}{x^{2}+x-12}\). We need to identify the values of x for which this function is defined.

Step 2: Identify the values of x that make the denominator zero

A rational function is undefined when the denominator is equal to zero. So we will solve the equation \(x^{2}+x-12=0\) to find these values. Factoring the equation, we get \((x-3)(x+4) = 0\). So, x = 3 and x = -4 makes the denominator zero.

Step 3: Identify the domain of the function

The domain of the given function is all real numbers except for the values that make the denominator zero. So the domain of the function \(g(x)\) is \(x \in R, x \neq 3, x \neq -4\) (where ‘R’ stands for all real numbers).

Key Concepts

Rational Functions

Rational functions are a type of function in mathematics where one polynomial is divided by another. Essentially, they are expressed as the ratio of two polynomials. A polynomial is an expression that consists of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, a rational function may look like this:

• \(f(x) = \frac{p(x)}{q(x)}\)

where \(p(x)\) and \(q(x)\) are polynomials, and \(q(x) eq 0\). The trick with rational functions is understanding where they are not defined. This is specifically where the denominator, \(q(x)\), equals zero because division by zero is undefined in mathematics. When setting the denominator equal to zero and solving for the variable, those x-values must be excluded from the domain of the function. Rational functions are fundamental in various mathematical fields due to their properties and the way they model real-world situations.

Factoring Quadratic Equations

Factoring quadratic equations is a method used to simplify and solve equations that are in the form of a quadratic. A typical quadratic equation looks like \[ ax^2 + bx + c = 0 \] where \(a\), \(b\), and \(c\) are constants. Factoring involves rewriting the quadratic as a product of two binomials, like \[ (x + m)(x + n) = 0 \] Here, the values of \(m\) and \(n\) are such that

• their sum is \(b\)
• their product is \(c\)

Factoring is useful because it allows us to find the roots of the equation easily. Once the equation is factored, you can set each binomial equal to zero and solve for \(x\). This gives you the x-values where the original quadratic equation is zero. These values are crucial when working with rational functions like in this exercise because they reveal potential restrictions on the domain by making the denominator zero.

Real Numbers

The set of real numbers is vast and encompasses various types of numbers. Real numbers include all the numbers that can be found on the number line. This consists of different categories such as:

• Positive integers (1, 2, 3, ...)
• Negative integers (-1, -2, -3, ...)
• Rational numbers (fractions like \(\frac{1}{2}\), \(\frac{2}{3}\))
• Irrational numbers (\(\pi\), \(\sqrt{2}\), which can't be expressed as simple fractions)

Real numbers are essential in mathematics because they help in counting, measuring and tagging. For example, when determining the domain of a function, we often refer to all possible x-values the function can take. In the case of rational functions, like the one in our exercise, the domain is defined as all real numbers except those that make the denominator zero. This concept ensures the function is mathematically valid and avoids undefined scenarios such as division by zero.