Math Problem Statement

find its domain

Solution

To find the domain of each expression, we need to determine the values of xx for which the expressions are defined. An expression is typically undefined where there is division by zero or any other operation that is not mathematically valid.

Problem 3: Finding the Domain

The simplified expression from Problem 3 is:

3x+7x2+2x1\frac{3x + 7}{x^2 + 2x - 1}

This expression is undefined when the denominator is zero. So, we need to find the values of xx that make x2+2x1=0x^2 + 2x - 1 = 0.

Step 1: Solve for xx in the Denominator

Set the denominator equal to zero:

x2+2x1=0x^2 + 2x - 1 = 0

Solve this quadratic equation using the quadratic formula, x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=1a = 1, b=2b = 2, and c=1c = -1.

x=2±2241(1)21x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}

x=2±4+42x = \frac{-2 \pm \sqrt{4 + 4}}{2}

x=2±82x = \frac{-2 \pm \sqrt{8}}{2}

x=2±222x = \frac{-2 \pm 2\sqrt{2}}{2}

x=1±2x = -1 \pm \sqrt{2}

The two values that make the denominator zero are x=1+2x = -1 + \sqrt{2} and x=12x = -1 - \sqrt{2}.

Step 2: Domain of Problem 3

The domain of the expression 3x+7x2+2x1\frac{3x + 7}{x^2 + 2x - 1} is all real numbers except x=1+2x = -1 + \sqrt{2} and x=12x = -1 - \sqrt{2}.

So, the domain is:

xR,x1+2,x12\boxed{x \in \mathbb{R}, \, x \neq -1 + \sqrt{2}, \, x \neq -1 - \sqrt{2}}

Problem 4: Finding the Domain

The simplified expression from Problem 4 is:

2x+3x+2\frac{2x + 3}{x + 2}

This expression is undefined when the denominator is zero. So, we need to find the value of xx that makes x+2=0x + 2 = 0.

Step 1: Solve for xx in the Denominator

Set the denominator equal to zero:

x+2=0x + 2 = 0

Solve for xx:

x=2x = -2

Step 2: Domain of Problem 4

The domain of the expression 2x+3x+2\frac{2x + 3}{x + 2} is all real numbers except x=2x = -2.

So, the domain is:

xR,x2\boxed{x \in \mathbb{R}, \, x \neq -2}

Summary of Domains

  • Problem 3 Domain: xR,x1+2,x12x \in \mathbb{R}, \, x \neq -1 + \sqrt{2}, \, x \neq -1 - \sqrt{2}
  • Problem 4 Domain: xR,x2x \in \mathbb{R}, \, x \neq -2

Would you like further explanations or have any specific questions? Here are some related questions to explore:

  1. How do we determine the domain of a function involving square roots?
  2. Why is division by zero undefined in mathematics?
  3. What is the quadratic formula, and when is it used?
  4. How do we solve inequalities to find the domain of an expression?
  5. Are there different methods for finding the domain of a more complex function?

Tip: Always check for values that make the denominator zero or cause other undefined operations when finding the domain of a rational expression.

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Math Problem Analysis

Mathematical Concepts

Domain of a Function
Rational Expressions
Quadratic Equations

Formulas

Quadratic formula

Theorems

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Suitable Grade Level

Grades 10-12