Find the domain of the function f(x) = x^2 + 2x + 1x^2 - 8x + 12.

Find the domain of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x…

Question 12 of 33intermediate🔧 ApplyShort Answer2 marks

Find the domain of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$ .

Correct Answer

To find the domain of the function $f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}$ , we must ensure that the denominator is not equal to zero, as division by zero is undefined.

Setting the denominator to zero, we solve $x^2 - 8x + 12 = 0$ . Factoring this quadratic expression, we get $(x - 6)(x - 2) = 0$ , which gives $x = 6$ and $x = 2$ .

Therefore, the function is defined for all real numbers except at $x = 2$ and $x = 6$ . Hence, the domain of $f$ is $R - \{2, 6\}$ .

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 3 | (Chapter: 17)

For More Understanding

Explanation

The solution follows the method demonstrated in Example 21 of the provided context. For a rational function, the domain includes all real numbers except those that make the denominator zero. By factorizing the denominator $x^2 - 8x + 12$ , we identify the specific values to exclude.

Solution Steps

Step 1: Identify the denominator of the function: $x^2 - 8x + 12$ .
Step 2: Set the denominator equal to zero to find undefined points: $x^2 - 8x + 12 = 0$ .
Step 3: Factorize the equation: $(x - 6)(x - 2) = 0$ .
Step 4: Solve for $x$ to get $x = 6$ and $x = 2$ .
Step 5: State the domain as the set of real numbers excluding these values: $R - \{2, 6\}$ .

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