Question
Decide whether each relation defines \( y \) as a function of \( x \). Give the domain and range. \( y=\frac{3}{x-2} \) Does this relation give a function? No Yes
Ask by Cervantes Cervantes. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
This relation defines a function.
Domain: All real numbers except 2.
Range: All real numbers except 0.
Solution
The relation y = 3/(x - 2) is indeed a function because for every x-value in its domain, there is exactly one corresponding y-value.
Domain:
x cannot be 2 because that would make the denominator zero. Therefore, the domain is all real numbers except 2:
Domain = {x ∈ ℝ | x ≠ 2} or (-∞, 2) ∪ (2, ∞).
Range:
Since y = 3/(x - 2) can never be 0 (because 3 divided by any nonzero number will never equal 0), the range is all real numbers except 0:
Range = {y ∈ ℝ | y ≠ 0}.
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Simplify this solution The Deep Dive
Yes, the relation \( y = \frac{3}{x-2} \) defines \( y \) as a function of \( x \). For every input \( x \) (except for \( x = 2 \), where the function is undefined), there is exactly one output \( y \). The domain of this function is all real numbers except \( x = 2 \), which can be expressed as \( (-\infty, 2) \cup (2, \infty) \). The range is also all real numbers except \( y = 0 \), since the function approaches \( 0 \) but never actually reaches it, making the range \( (-\infty, 0) \cup (0, \infty) \).
