Video: Evaluating a an Evaluation of a Point of a Piecewise Function

Consider the function 𝑓(π‘₯) = π‘₯ + 4 if π‘₯ > 4, 𝑓(π‘₯) = 2π‘₯ if βˆ’1 ≀ π‘₯ ≀ 4, and 𝑓(π‘₯) = βˆ’3 if π‘₯ < βˆ’1. Find 𝑓[𝑓(2)].

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Video Transcript

Consider the function 𝑓 of π‘₯ is equal to π‘₯ plus four if π‘₯ is greater than four, two π‘₯ if π‘₯ is greater than or equal to negative one and less than or equal to four, and negative three if π‘₯ is less than negative one. Find 𝑓 of 𝑓 of two.

𝑓 of 𝑓 of two is a composite function. It’s a function of a function. We’re going to begin by looking at the inner function first, so we’re going to begin by thinking about 𝑓 of two. Now, our 𝑓 of π‘₯ is a piecewise function, and it’s defined by different functions on different intervals of π‘₯. We’re told that when π‘₯ is greater than four to use the function π‘₯ plus four. When π‘₯ is between and including negative one and four, we use the function two π‘₯. And when π‘₯ is less than negative one, we use the function 𝑓 of π‘₯ equals negative three. Two, of course, lies between negative one and four, and so we’re going to use the second part of our function. That is, when π‘₯ is equal to two, 𝑓 of π‘₯ is equal to the function two π‘₯.

And so, 𝑓 of two is found by substituting two into this equation. We get two times two, which is four. So we found 𝑓 of two; it’s four. If we replace 𝑓 of two with its value of four, we see that we now need to evaluate 𝑓 of four. And we need to be really careful here. We’re actually still using this second part of the function. And this is because we only use the first part of the function when π‘₯ is strictly greater than four. When it’s less than or equal to four, we use the function two π‘₯. And so once again, we substitute our value of π‘₯ into the function 𝑓 of π‘₯ equals two π‘₯, so it’s two times four which is equal to eight. Given our piecewise function, 𝑓 of 𝑓 of two is eight.

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