Question

What are the domain and range of the function f(x)=-3(x-5)2 +4?

Answer 1

Answers:

Domain is $(-\infty, \infty)$

Range is $(-\infty, 4]$

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Explanation:

We can replace x with any real number we want. We don't have any restrictions to worry about since there are no division by zero issues for instance. Also, there isn't any issues of things like taking the square root of a negative number.

Therefore, the domain is the set of all real numbers which translates to the interval notation of $(-\infty, \infty)$

This interval notation can be thought of as $-\infty < x < \infty$

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The range on the other hand isn't the set of all real numbers. It might help to graph this parabola (see below). You should see that the highest point occurs at the vertex (5, 4). This then tells us that the largest y can get is y = 4.

In other words, y = 4 or y can be smaller than this.

In symbols, we would say the range is $y \le 4$ and that translates to the interval notation of $(-\infty, 4]$

The curved parenthesis always goes with either infinity. The square bracket says "include the endpoint 4 as part of the interval".