Question Video: Evaluating a Piecewise Function at a Given Point Mathematics

Video Transcript

Given that the function 𝑓 of 𝑥 is equal to six 𝑥 minus two if 𝑥 is less than negative six, negative nine 𝑥 squared minus one if 𝑥 is greater than or equal to negative six and less than or equal to eight, and negative five 𝑥 cubed plus four if 𝑥 is greater than eight, find the value of 𝑓 of four.

We see that 𝑓 of 𝑥 is a piecewise function, and it’s defined by three separate functions. When 𝑥 is less than negative six, we’re going to use the function 𝑓 of 𝑥 equals six 𝑥 minus two. When 𝑥 is between negative six and eight and including those values, we use the function negative nine 𝑥 squared minus one. And when 𝑥 is greater than eight, we use the function 𝑓 of 𝑥 is negative five 𝑥 cubed plus four. Now we want to find the value of 𝑓 of four. And so we need to make sure that we correctly select the function that we need to use when 𝑥 is equal to four. Well, four is between negative six and eight. So we’re going to use this part of the function: 𝑓 of 𝑥 is negative nine 𝑥 squared minus one.

And so, 𝑓 of four is found by substituting 𝑥 equals four into this function. It’s negative nine times four squared minus one. Now, of course, the order of operations, which is sometimes abbreviated to PEMDAS or BIDMAS, tells us to begin by working out the value of the number being raised to some exponent. So in this case, we begin by working out four squared. That’s four times four which is 16. And so our calculation becomes negative nine times 16 minus one. We then perform the multiplication part of this calculation, remembering that a negative multiplied by a positive is a negative. We get negative 144 minus one. Negative 144 minus one is negative 145. And so, given the piecewise function 𝑓 of 𝑥, we see that 𝑓 of four is negative 145.