To determine if the relation [tex]\( y = \sqrt{x^2 - 10} \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we need to understand the requirements for a relation to be classified as a function. Specifically, for each value of [tex]\( x \)[/tex] in the domain, there must be exactly one corresponding value of [tex]\( y \)[/tex].
1. Function Definition: By definition, a function is a relation where each input (in this case [tex]\( x \)[/tex]) has a unique output (in this case [tex]\( y \)[/tex]).
2. Analyzing the Given Equation: Given the relation [tex]\( y = \sqrt{x^2 - 10} \)[/tex]:
- This equation involves a square root, which generally gives a non-negative result.
- However, since square roots can represent both the positive and negative roots, we need to consider both [tex]\( y = \sqrt{x^2 - 10} \)[/tex] and [tex]\( y = -\sqrt{x^2 - 10} \)[/tex].
3. Domain Restrictions: For [tex]\( y \)[/tex] to be a real number, [tex]\( x^2 - 10 \geq 0 \)[/tex]. Simplifying this inequality, we get [tex]\( x^2 \geq 10 \)[/tex] which implies [tex]\( x \geq \sqrt{10} \)[/tex] or [tex]\( x \leq -\sqrt{10} \)[/tex]. The domain of [tex]\( x \)[/tex] thus includes all values such that [tex]\( x \geq \sqrt{10} \)[/tex] or [tex]\( x \leq -\sqrt{10} \)[/tex].
4. Example Analysis: Let's examine specific values of [tex]\( x \)[/tex] within the domain.
- If [tex]\( x = 4 \)[/tex], we calculate:
- [tex]\( y_1 = \sqrt{4^2 - 10} = \sqrt{16 - 10} = \sqrt{6} \approx 2.449 \)[/tex]
- [tex]\( y_2 = -\sqrt{4^2 - 10} = -\sqrt{16 - 10} = -\sqrt{6} \approx -2.449 \)[/tex]
From this example, [tex]\( x = 4 \)[/tex] corresponds to two different [tex]\( y \)[/tex]-values: [tex]\( \sqrt{6} \)[/tex] and [tex]\( -\sqrt{6} \)[/tex]. This indicates that the given relation is not a function of [tex]\( x \)[/tex] because a function would not map one [tex]\( x \)[/tex] to multiple [tex]\( y \)[/tex]-values.
5. Conclusion: Since we found an instance where a single [tex]\( x \)[/tex]-value corresponds to two different [tex]\( y \)[/tex]-values, this demonstrates the relation does not satisfy the definition of a function.
Therefore, the correct choice is:
A. False, because each member of the domain does not correspond to exactly one member of the range for the given equation. For example, [tex]\( x = 4 \)[/tex] corresponds to two [tex]\( y \)[/tex]-values, [tex]\( \sqrt{6} \)[/tex] and [tex]\( -\sqrt{6} \)[/tex].
