Question

station #1 - analyzing functions
think about the following function: $\frac{-3}{sqrt{5 - x}}$
what do you think the domain is?
hint for domain: there is a fraction and a square root, so what do these things mean for the values of x?
what do you think the range is?
hint for range: as x gets closer and closer to 5, what happens to the y - values? what happens when the values of x get further from 5?
now graph the function to see what it looks like on your calculator. how does it compare? why do you think the graph looks like this?

Explanation:

Step1: Determine the domain

For the function $y = \frac{- 3}{\sqrt{5 - x}}$, the expression under the square - root must be non - negative in the real number system, and since it's in the denominator, $5−x>0$. Solving $5−x>0$ gives $x < 5$. So the domain is $x\in(-\infty,5)$.

Step2: Determine the range

As $x$ approaches $5$ from the left ($x\to5^{-}$), $\sqrt{5 - x}\to0^{+}$, and $y=\frac{-3}{\sqrt{5 - x}}\to-\infty$. As $x$ goes to $-\infty$, $\sqrt{5 - x}\to+\infty$, and $y = \frac{-3}{\sqrt{5 - x}}\to0^{-}$. So the range is $y\in(-\infty,0)$.

Step3: Analyze the graph

The function has a vertical asymptote at $x = 5$ (because the denominator approaches $0$ as $x\to5$) and the function is always negative. As $x$ decreases from values less than $5$, the function values get closer to $0$ but are always negative.

Answer:

Domain: $x\in(-\infty,5)$; Range: $y\in(-\infty,0)$; The graph has a vertical asymptote at $x = 5$ and lies entirely below the $x$ - axis.