Question

How do you find the domain and range of `y=x^2+3x+1`?

Answer 1

`"D":{x inRR}`
`"R":{y inRR|y>=-1.25}`.

Explanation:

The domain and range are a set of all the possible values that a function can have.

Domain refers to the `x`-coordinate, and range refers to the `y`-coordinate.

For a parabola, no matter the values, the domain will always be `"D":{x inRR}` (unless context is given).

As for range, the range is dependent on the `c`-value of the equation (only in vertex form).

This is because, if the `c`-value (again, only in vertex form) is `1`, then the parabola is moved up `1` units. Meaning any value below `1` is inadmissible.

Unfortunately, in this case (standard form), the `c`-value refers to the `y`-intercept. We can convert the equation to vertex form, or we can graph it and examine the parabola. We'll do that.

graph{x^2 + 3x + 1 [-3.895, 3.9, -1.948, 1.947]}

As you can see, the domain can be any `x`-value, while the range can only be values equal to or above the vertex's `y`-coordinate, `-1.25`.

Therefore, the range is `"R":{y inRR|y>=-1.25}`.

Hope this helps :)