https://file.aiquickdraw.com/mathbot/file/1721174939799vyyssflu.jpg

Identify the Domain of each function. Use interval notation.

To determine the domain of the function \( f(x) = \sqrt{x^2 - 5x - 36} \), we need to ensure that the expression inside the square root is non-negative, since the square root function is only defined for non-negative numbers.

The expression inside the square root is \( x^2 - 5x - 36 \). We set this expression greater than or equal to zero to find the values of \( x \) that satisfy this condition:

\[ x^2 - 5x - 36 \geq 0 \]

First, we solve the quadratic equation \( x^2 - 5x - 36 = 0 \):

To factorize \( x^2 - 5x - 36 \), we find two numbers that multiply to \(-36\) and add up to \(-5\). These numbers are \(-9\) and \(4\):

\[ x^2 - 5x - 36 = (x - 9)(x + 4) \]

So, the quadratic equation can be written as:

\[ (x - 9)(x + 4) = 0 \]

The solutions to this equation are:

\[ x - 9 = 0 \implies x = 9 \]
\[ x + 4 = 0 \implies x = -4 \]

These solutions divide the real number line into three intervals: \((-\infty, -4)\), \((-4, 9)\), and \((9, \infty)\).

Next, we determine the sign of \( x^2 - 5x - 36 \) in each interval:

1. For \( x \in (-\infty, -4) \):
   Choose a test point, \( x = -5 \):
   \[ (-5)^2 - 5(-5) - 36 = 25 + 25 - 36 = 14 \]
   Since 14 is positive, \( x^2 - 5x - 36 \geq 0 \) in this interval.

2. For \( x \in (-4, 9) \):
   Choose a test point, \( x = 0 \):
   \[ 0^2 - 5(0) - 36 = -36 \]
   Since -36 is negative, \( x^2 - 5x - 36 < 0 \) in this interval.

3. For \( x \in (9, \infty) \):
   Choose a test point, \( x = 10 \):
   \[ 10^2 - 5(10) - 36 = 100 - 50 - 36 = 14 \]
   Since 14 is positive, \( x^2 - 5x - 36 \geq 0 \) in this interval.

Therefore, the domain of \( f(x) \) is where \( x^2 - 5x - 36 \geq 0 \):

\[ x \in (-\infty, -4] \cup [9, \infty) \]

In interval notation, the domain is:

\[ (-\infty, -4] \cup [9, \infty) \]

Would you like more details or have any questions?

Here are 5 numbered questions for you to ask:
1. How do you find the range of this function?
2. Can you explain how to solve quadratic inequalities?
3. What is the significance of the test points in finding intervals?
4. How do you find the domain of other types of functions?
5. Can you show how to graph this function?

**Tip:** When solving quadratic inequalities, always check the sign of the quadratic expression in each interval determined by its roots.

Learn how to identify the domain of a function using interval notation with a detailed step-by-step solution. This problem involves analyzing the quadratic function f(x) = sqrt(x^2 - 5x - 36) to determine where the function is defined. Suitable for high school students studying functions and quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation