Question

Which of the following could be an example of a function with a range (-∞, α) and a domain [b, ∞) where a > 0 and b>0?

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Answer 1

Answer: Choice D

Step-by-step explanation:

Choices A and B are ruled out because they are cube root functions, which have a domain and range of "all real numbers".

Choice C is ruled out because the domain is
`[a, \infty)`

To determine this domain, we set the radicand (aka stuff under the square root) to be greater than or equal to zero.

`x-a \ge 0 \ \ \text{ solves to } \ \ x \ge a`

`x \ge a \ \ \text{ turns into the interval notation } \ \ [a, \infty)`

The square bracket is used to include the value 'a' as part of the interval. This is the interval from x = a to positive infinity.

The domain of choice D is
`[b, \infty)` found using similar steps as shown above.

The range is
`(-\infty, a]` since this square root function is decreasing, due to the negative out front, which means y = a the largest output possible. I recommend graphing it out using a tool like desmos or geogebra. These tools offer a way to use parameters.