Video Transcript
Consider the function 𝑓 of 𝑥 is
equal to 𝑥 plus four if 𝑥 is greater than four, two 𝑥 if 𝑥 is greater than or
equal to negative one and less than or equal to four, and negative three if 𝑥 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 𝑥 is a piecewise
function, and it’s defined by different functions on different intervals of 𝑥. We’re told that when 𝑥 is greater
than four to use the function 𝑥 plus four. When 𝑥 is between and including
negative one and four, we use the function two 𝑥. And when 𝑥 is less than negative
one, we use the function 𝑓 of 𝑥 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 𝑥 is equal to two,
𝑓 of 𝑥 is equal to the function two 𝑥.
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 𝑥 is strictly greater than four. When it’s less than or equal to
four, we use the function two 𝑥. And so once again, we substitute
our value of 𝑥 into the function 𝑓 of 𝑥 equals two 𝑥, so it’s two times four
which is equal to eight. Given our piecewise function, 𝑓 of
𝑓 of two is eight.
