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To determine for which values of [tex]\( x \)[/tex] the function [tex]\( f(x) = \sqrt{x^2 - 2x - 8} \)[/tex] is undefined, we need to consider the expression under the square root, [tex]\( x^2 - 2x - 8 \)[/tex], and identify when it becomes negative. The function [tex]\( f \)[/tex] is undefined for values of [tex]\( x \)[/tex] where the expression under the square root is negative, as the square root of a negative number is not a real number.
Let’s solve the inequality [tex]\( x^2 - 2x - 8 < 0 \)[/tex] to find the intervals where the function is undefined.
1. Find the roots of the quadratic equation:
To find when [tex]\( x^2 - 2x - 8 \)[/tex] is zero, solve the equation:
[tex]\[ x^2 - 2x - 8 = 0 \][/tex]
Factoring the quadratic equation:
[tex]\[ (x - 4)(x + 2) = 0 \][/tex]
This gives us the roots:
[tex]\[ x = 4 \quad \text{and} \quad x = -2 \][/tex]
2. Determine the sign of the quadratic expression:
The roots divide the number line into three intervals: [tex]\( (-\infty, -2) \)[/tex], [tex]\( (-2, 4) \)[/tex], and [tex]\( (4, \infty) \)[/tex]. We need to test the sign of the expression [tex]\( x^2 - 2x - 8 \)[/tex] in each of these intervals.
- For [tex]\( x \in (-\infty, -2) \)[/tex]:
Choose a test point, say [tex]\( x = -3 \)[/tex]:
[tex]\[ (-3)^2 - 2(-3) - 8 = 9 + 6 - 8 = 7 \quad (\text{positive}) \][/tex]
- For [tex]\( x \in (-2, 4) \)[/tex]:
Choose a test point, say [tex]\( x = 0 \)[/tex]:
[tex]\[ 0^2 - 2(0) - 8 = -8 \quad (\text{negative}) \][/tex]
- For [tex]\( x \in (4, \infty) \)[/tex]:
Choose a test point, say [tex]\( x = 5 \)[/tex]:
[tex]\[ 5^2 - 2(5) - 8 = 25 - 10 - 8 = 7 \quad (\text{positive}) \][/tex]
3. Identify the interval where [tex]\( x^2 - 2x - 8 \)[/tex] is negative:
From the test points, we see that the expression [tex]\( x^2 - 2x - 8 \)[/tex] is negative in the interval [tex]\( (-2, 4) \)[/tex].
Therefore, the function [tex]\( f(x) = \sqrt{x^2 - 2x - 8} \)[/tex] is undefined for [tex]\( x \)[/tex] in the interval [tex]\( (-2, 4) \)[/tex]. The critical points where the expression under the square root exactly equals zero are [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], but at these points, the function is zero, not undefined. Thus, [tex]\( f(x) \)[/tex] is undefined strictly in the open interval [tex]\( (-2, 4) \)[/tex].
Let’s solve the inequality [tex]\( x^2 - 2x - 8 < 0 \)[/tex] to find the intervals where the function is undefined.
1. Find the roots of the quadratic equation:
To find when [tex]\( x^2 - 2x - 8 \)[/tex] is zero, solve the equation:
[tex]\[ x^2 - 2x - 8 = 0 \][/tex]
Factoring the quadratic equation:
[tex]\[ (x - 4)(x + 2) = 0 \][/tex]
This gives us the roots:
[tex]\[ x = 4 \quad \text{and} \quad x = -2 \][/tex]
2. Determine the sign of the quadratic expression:
The roots divide the number line into three intervals: [tex]\( (-\infty, -2) \)[/tex], [tex]\( (-2, 4) \)[/tex], and [tex]\( (4, \infty) \)[/tex]. We need to test the sign of the expression [tex]\( x^2 - 2x - 8 \)[/tex] in each of these intervals.
- For [tex]\( x \in (-\infty, -2) \)[/tex]:
Choose a test point, say [tex]\( x = -3 \)[/tex]:
[tex]\[ (-3)^2 - 2(-3) - 8 = 9 + 6 - 8 = 7 \quad (\text{positive}) \][/tex]
- For [tex]\( x \in (-2, 4) \)[/tex]:
Choose a test point, say [tex]\( x = 0 \)[/tex]:
[tex]\[ 0^2 - 2(0) - 8 = -8 \quad (\text{negative}) \][/tex]
- For [tex]\( x \in (4, \infty) \)[/tex]:
Choose a test point, say [tex]\( x = 5 \)[/tex]:
[tex]\[ 5^2 - 2(5) - 8 = 25 - 10 - 8 = 7 \quad (\text{positive}) \][/tex]
3. Identify the interval where [tex]\( x^2 - 2x - 8 \)[/tex] is negative:
From the test points, we see that the expression [tex]\( x^2 - 2x - 8 \)[/tex] is negative in the interval [tex]\( (-2, 4) \)[/tex].
Therefore, the function [tex]\( f(x) = \sqrt{x^2 - 2x - 8} \)[/tex] is undefined for [tex]\( x \)[/tex] in the interval [tex]\( (-2, 4) \)[/tex]. The critical points where the expression under the square root exactly equals zero are [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], but at these points, the function is zero, not undefined. Thus, [tex]\( f(x) \)[/tex] is undefined strictly in the open interval [tex]\( (-2, 4) \)[/tex].
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