Question

Find the domain of \(h(x)=\frac{x+2}{x^{2}-5 x-14}\).

Short answer

The domain is all real numbers except -2 and 7.

Step-by-step solution

- Identify the Denominator

To find the domain of the function, start by identifying the denominator of the fraction. In this case, the denominator is: \[x^2 - 5x - 14\]

- Set the Denominator Equal to Zero

Next, set the denominator equal to zero to find the values of x that make the function undefined: \[x^2 - 5x - 14 = 0\]

- Solve the Quadratic Equation

Solve the quadratic equation for x. The equation is a standard quadratic equation of the form \(ax^2 + bx + c = 0\). To solve it, factorize it: \[x^2 - 5x - 14 = (x - 7)(x + 2) = 0\]

- Find the Values that Make the Denominator Zero

Set each factor equal to zero and solve for x:\[(x - 7) = 0 \quad \Rightarrow \quad x = 7\] \[(x + 2) = 0 \quad \Rightarrow \quad x = -2\]

- Determine the Domain

The domain of the function consists of all real numbers except those that make the denominator zero. Therefore, the domain is:\[x \in \mathbb{R} \setminus \{-2, 7\}\]

Key concepts

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. A key aspect of understanding rational functions is identifying the values where they become undefined.

This happens when the denominator is zero because division by zero is not allowed in mathematics. For example, with the function you're working with, ewline ewline The domain is determined by identifying where the denominator is not equal to zero. ewline The function in your question, ewline ewline Let's dive into rational functions more to simplify your understanding.

• Components of Rational Functions: Understand the expressions above and below the fraction line.
• Undefined Values: Identify problematic values where the denominator equals zero.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, which can be generally written as: ewline ewline The form speaks about its degree and its characteristics. The solutions or roots of quadratic equations are the values of x that satisfy the equation. ewline Quadratic equations are used frequently in various fields.
To solve the quadratic equation, you can apply the factoring method, the quadratic formula, or completing the square method. Let's see a brief overview:

• Factoring: This involves rewriting the equation in a product form and setting each factor equal to zero to find solutions.
• Quadratic Formula: You can also use a standard formula to find the roots when factoring is not straightforward.
• Completing the Square: This is a method where you seek to form a perfect square trinomial, which is useful for more complex equations.

Using the factoring method in your case, we rewrote the equation in a form that made it easy to solve for x.

Factoring

Factoring is a technique where you express a polynomial as a product of simpler polynomials. It is an essential skill in algebra, especially when working on quadratic equations.

To understand factoring better:

• Identify Common Factors: Look for numbers or variables common in each term of the polynomial. Pull these out first.
• Recognize Patterns: Be familiar with patterns such as the difference of squares or perfect square trinomials.
• Quadratic Polynomials: Specifically for quadratic polynomials, look for factors that multiply to give you the 'ac' term and add to give you the 'b' term in the general form ax^2 + bx + c.

In the solution provided, we factored the quadratic denominator to find the values making the function undefined. This is a powerful method to simplify and solve equations, making it easier to determine the domain of rational functions.