Question

Determine whether each relation defines y as a function of x. Give the domain and range. y=2/(x-3)

Accepted Answer

Identify the given relation: $$y = \frac{2}{x - 3}. $$Determine if $$y is a $$function of $$x$$: For each value of $$x ($$except where the expression is undefined), there should be exactly one corresponding value of $$y. $$Since the expression is a fraction with $$x in $$the denominator, check where the denominator equals zero. Find the values of $$x$$ that make the denominator zero by solving $$x - 3 = 0$$, which gives $$x = 3. $$This value is excluded from the domain because division by zero is undefined. State the domain: All real numbers except $$x = 3$$, which can be written as $$\{x \in \mathbb{R} \mid x 
eq 3\}. $$Determine the range: Since $$y = \frac{2}{x - 3}$$ can take any real value except possibly some value that $$y$$ cannot equal, analyze if there are any restrictions on $$y. $$Because the function is a rational function with a vertical asymptote at $$x=3$$ and no horizontal asymptote restricting $$y$$, the range is all real numbers except $$y = 0 ($$since $$\frac{2}{x-3} = 0$$ has no solution).

Determine whether each relation defines y as a function of x. Give the domain and range. y=2/(x-3)

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Key Concepts

Definition of a Function

Domain of a Function

Range of a Function