Question

drag the intervals to the correct location on the image. each interval can be used more than once, but not all intervals will be used. the graph of rational function h displays these key features: vertical asymptote at \\(x = 4\\), horizontal asymptote at \\(y = -3\\), hole at \\((5, -2)\\). what are the domain and the range of function h? domain: \\(\square \cup \square \cup \square\\), range: \\(\square \cup \square \cup \square\\). intervals: \\((5, \infty)\\), \\((-3, -2)\\), \\((-3, 4)\\), \\((-\infty, -3)\\), \\((-\infty, 4)\\), \\((4, 5)\\), \\((-2, 5)\\), \\((-2, \infty)\\)

Explanation:

Step1: Find domain restrictions

A rational function's domain excludes values that make the denominator zero. There is a vertical asymptote at $x=4$ (so $x=4$ is excluded) and a hole at $(5,-2)$ (so $x=5$ is excluded).

Step2: Write domain intervals

Split the real numbers into intervals excluding $x=4$ and $x=5$:
$(-\infty, 4)$, $(4, 5)$, $(5, \infty)$

Step3: Find range restrictions

The horizontal asymptote at $y=-3$ means $y=-3$ is excluded. The hole at $(5,-2)$ means $y=-2$ is also excluded (the function never takes this value).

Step4: Write range intervals

Split the real numbers into intervals excluding $y=-3$ and $y=-2$:
$(-\infty, -3)$, $(-3, -2)$, $(-2, \infty)$

Answer:

Domain: $\boldsymbol{(-\infty, 4) \cup (4, 5) \cup (5, \infty)}$
Range: $\boldsymbol{(-\infty, -3) \cup (-3, -2) \cup (-2, \infty)}$