Question

Analyze the graph of the function f(x) to complete the statement.

On a coordinate plane, a curved line, labeled f of x, with a minimum value of (0, negative 3) and a maximum value of (negative 2.4, 17), crosses the x-axis at (negative 3, 0), (negative 1.1, 0), and (0.9, 0), and crosses the y-axis at (0, negative 3).

f(x)<0 over and what other interval?
( Can someone explain this IN DEPTH please?)

Answer 1

f(x) is the same as y, so we can say y = f(x)

Writing f(x) < 0 means we want to find when y < 0.

Visually, we are looking at the graph when the curve is below the horizontal x axis.

This is the portion in red that I have marked in the diagram (see attached image below). I apologize for the numbers being blurry.

The left red portion is from negative infinity to -3. In terms of a compound inequality we write
`-\infty < x < -3` which in interval notation is
`(-\infty, -3)`. The curved parenthesis tells the reader to exclude both endpoints.

The right red portion is from x = -1.1 to x = 0.9, excluding both endpoints. So we say
`-1.1 < x < 0.9` which becomes the interval notation
`(-1.1, 0.9)`. This is not ordered pair notation even though it looks identical to it.

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Answer in interval notation:
`(-\infty, -3) \cup (-1.1, 0.9)`

The "U" means "set union" which glues together the two separate intervals. Basically it's saying "x is either in the interval (-infinity, -3) or it is in the interval (-1.1, 0.9)"