Video Transcript
Find the missing table values for
the piecewise function 𝑔 of 𝑥, which is equal to two to the power of 𝑥 if 𝑥 is
less than negative two, three to the power of 𝑥 if 𝑥 is greater than or equal to
negative two and less than three, or two to the power of 𝑥 if 𝑥 is greater than or
equal to three. And then we have a table with the
values of 𝑥, negative three, zero, and three.
Remember, when we have a function
defined by different functions depending on its value of 𝑥, we call it a piecewise
function. And according to our table, we’re
looking to find the value of 𝑔 of 𝑥 when 𝑥 is negative three. So that’s 𝑔 of negative three. We also want to find 𝑔 of zero and
𝑔 of three. And so we need to pay extra careful
attention to the part of the function we’re going to use for each value of 𝑥. Let’s begin with 𝑔 of negative
three. Here, 𝑥 is equal to negative
three. And so, since negative three is
less than negative two, we need to use the first part of our function, that is, two
to the power of 𝑥.
And so to find 𝑔 of negative
three, we’re going to substitute 𝑥 equals negative three into that part of the
function. And we get 𝑔 of negative three is
two to the power of negative three. And at this stage, we might recall
that a negative power tells us to find the reciprocal. So 𝑎 to the power of negative 𝑏,
for instance, is one over 𝑎 to the power of 𝑏. And this means then that two to the
power of negative three is one over two cubed, which is equal to one over eight. And so the first value in our table
is one-eighth. Let’s repeat this process for 𝑥
equals zero.
This time zero is between negative
two and three, so we’re going to use the second part of our function. And so, 𝑔 of zero is found by
substituting 𝑥 equals zero into the function 𝑔 of 𝑥 equals three to the power of
𝑥. So that’s three to the power of
zero. Now, of course, at this stage, we
might recall that anything to the power of zero is equal to one, so 𝑔 to the power
of zero is simply equal to one. And that’s the second value in our
table. Let’s repeat this process for the
third and final column in our table.
The third part of our piecewise
function is used when 𝑥 is greater than or equal to three. So we’re going to use this value
when 𝑥 is equal to three. And this means that 𝑔 of three is
two cubed, which is simply equal to eight. And so we pop eight in the final
part of our table. The missing table values for our
piecewise function 𝑔 of 𝑥 are one-eighth, one, and eight.
