Question 26 of 35intermediate🔧 ApplyShort Answer2 marks

Find the domain of the function f(x)=x2+2x+1x2−8x+12f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}.

Correct Answer

To find the domain of the function f(x)=x2+2x+1x2−8x+12f(x) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}, we need to determine the values of xx for which the function is defined.

The function is defined for all real numbers except where the denominator becomes zero. Factorizing the denominator: x2−8x+12=(x−6)(x−2)x^2 - 8x + 12 = (x-6)(x-2). The denominator equals zero when x=6x = 6 or x=2x = 2.

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

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 3 | (Page 17)
For More Understanding

Explanation

This question follows the same pattern as Example 21 in the textbook. The domain of a rational function is all real numbers except those that make the denominator zero. By factorizing the denominator x² - 8x + 12 = (x-6)(x-2), we find the values x = 6 and x = 2 make the denominator zero, so these must be excluded from the domain.

Solution Steps

  1. Step 1: Identify that the function is undefined when denominator equals zero

  2. Step 2: Factorize denominator: x² - 8x + 12 = (x-6)(x-2)

  3. Step 3: Find values where denominator is zero: x = 6 and x = 2

  4. Step 4: State domain as all real numbers except these values: R - {2, 6}

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