Question

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. 2 y = ——— x - 3

Accepted Answer

Identify the given relation: $$y = \frac{2}{x - 3}. $$This is a rational function where $$y$$ depends on $$x. $$Determine if $$y is a $$function of $$x$$: For each value of $$x ($$except where the denominator is zero), there is exactly one value of $$y. $$Since the denominator $$x - 3$$ cannot be zero, $$x 
eq 3. $$Therefore, $$y is a $$function of $$x. $$Find the domain: The domain consists of all real numbers except where the denominator is zero. Set the denominator equal to zero and solve for $$x$$: $$x - 3 = 0 \Rightarrow x = 3. So$$, the domain is all real numbers $$x$$ such that $$x 
eq 3. $$Find the range: Consider the values that $$y$$ can take. Since $$y = \frac{2}{x - 3}$$, $$y$$ can be any real number except where the function is undefined or does not reach. To find any restrictions, set $$y = 0$$ and solve for $$x$$: $$0 = \frac{2}{x - 3}$$, which has no solution. So, $$y$$ can never be zero. Hence, the range is all real numbers $$y$$ such that $$y 
eq 0. $$Summarize: The relation defines $$y as a $$function of $$x$$ with domain $$\{x \in \mathbb{R} : x 
eq 3\}$$ and range $$\{y \in \mathbb{R} : y 
eq 0\}.$$

Determine whether each relation defines y as a function of x. Give the domain and range. See Example 5. 2 y = ——— x - 3

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Concetti chiave

Definition of a Function

Domain of a Function

Range of a Function