Determine where the function 𝑓 of 𝑥 equals 𝑥 squared minus 400 is positive.
So if we want to determine where the function 𝑓 of 𝑥 equals 𝑥 squared minus 400 is positive, then what we want to do is set up an inequality. And this inequality is 𝑥 squared minus 400 is greater than zero. Well, if we want to solve an inequality like this, then the first thing we want to do is set it equal to zero. And we want to do that because what we’re trying to find is the critical values. And that’s because we’ve got a quadratic inequality.
Well, if we’ve got 𝑥 squared minus 400 is equal to zero, well, then there’s a couple of ways that we could find the critical values. First of all, we can find the solutions by factoring. So if we factor, we’ve got the difference of two squares. So as we can see that 400 is 20 squared, then what we’re gonna have is 𝑥 plus 20 multiplied by 𝑥 minus 20. So therefore, from this, if we want to work out what the 𝑥-values are, we set each of our parentheses equal to zero. And we do that because if we have one of them is zero, then we multiply by the other parentheses. It doesn’t matter what the values are. It will give us our answer as zero.
So therefore, if we’ve got 𝑥 plus 20 equals zero, then that means we know that 𝑥 is equal to negative 20. And if we have 𝑥 minus 20 equals zero, then therefore we know that 𝑥 is gonna be equal to 20. Alternatively, what we could have done is we could’ve added 400 to each side of the equation. And we could’ve added 𝑥 squared equals 400 and then taken the square root to each side of the equation. And that would’ve given us the same answer because we’ve already got 𝑥 is equal to positive or negative 20.
Okay, great, now, we’ve got our critical values, what do we do? How can we determine where the function 𝑓 of 𝑥 equals 𝑥 squared minus 400 is positive? Well, to help us, what we can do is draw a sketch. And what I’ve done is drawn a sketch here. So I’ve shown that our function looks like a U-shaped parabola. And that’s because we’ve gone an 𝑥 squared term that’s positive. Also, I’ve shown that it crosses the 𝑥-axis at negative 20 and positive 20. And that’s because these are our critical values. What I’ve also shown is where it crosses the 𝑦-axis. This isn’t necessary generally. But I’ve just done it just to show you how. And that’s found by substituting in 𝑥 equals zero. And if we did that in our function, we’d get zero minus 400 equals 𝑓 of 𝑥. So therefore, 𝑓 of 𝑥 would be equal to negative 400.
So now, we can use this to help us solve our inequality because we’re looking for the region where our function 𝑥 squared minus 400 is greater than zero. So therefore, we want everything above the 𝑥-axis. And if we want to rewrite this using inequality notation, we can say that 𝑥 is less than negative 20 or 𝑥 is greater than 20. But if we show this using set notation, we can say that we’ve determined where the function 𝑓 of 𝑥 equals 𝑥 squared minus 400 is positive. And it’s positive with all the real numbers except. And we’ve put that with an ℝ and then a minus. And then, we’ve got the interval between negative 20 and 20. And we’re using square brackets to show that it includes negative 20 and 20. Because, as we showed in our inequality, these would not be values that are valid.
So we’ve now solved the problem and solved our inequality and given the answer in two forms.