Determine the sign of the function represented by the figure below in real numbers.
When we’re determining the sign of a function, we have three choices: positive, zero, or negative. When a function falls into the shaded areas in green, we call it positive. That is to say, when the output or the 𝑦-value of a function is positive, the sign of the function is positive. Likewise, the sign of a function is negative, if it falls in the areas shaded red. If the 𝑦-value or the output of your function is negative, the sign of the function is negative. We also have a third option, when the sign of the function is zero. That is the case when the output or the 𝑦-value of your function is zero, and when your function lands on the 𝑥-axis on a graph.
Let’s look back at the original problem we were asked about. Where would this function be positive? Where are the 𝑦-values positive? Okay. We have this space from negative one to three where the 𝑦-values are all positive. For zero, we have two places where this function crosses the 𝑥-axis, at negative one and at three. Our function falls in the negatives everywhere to the left of negative one. It’s also negative everywhere that’s to the right of positive three. Now, how should we write this? How should we represent this?
We’ll start with the positive case. When 𝑥 falls between negative one and three, it’s positive. This is how we would write it. We use these brackets facing outward to represent 𝑥 that does not include negative one and three. It’s not equal to negative one and three, only between those two places. Another way to represent that looks like this. You might be more familiar with the parentheses that means does not include negative one and three. But, both these parentheses and the outward facing brackets mean the same thing.
And where is our function equal to zero? When 𝑥 equals negative one and three, the sign of our function is zero.
And where is our 𝑥-value negative? What we wanna say here, is that it’s negative everywhere. It’s not positive or zero. We want to say, it’s negative everywhere else. 𝑥 is negative when it is any number, or all real numbers with the exception of, subtracted from, the portion from negative one to three. In other words, any time it’s not positive or zero, our function is negative. All reals minus the piece from negative one to three.