Explore a diverse range of topics and get answers from knowledgeable individuals on IDNLearn.com. Our platform offers comprehensive and accurate responses to help you make informed decisions on any topic.
Sagot :
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].
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.