Video Transcript
Use the table to evaluate 𝑓 of 𝑔 of two and 𝑔 of 𝑓 of one.
Let’s have a look at this table and understand what it is telling us first. There are four values of 𝑥: one, two, three, and four. The next column tells us the values of the function 𝑓 for each of these values of 𝑥. For example, 𝑓 of three is equal to nine. The final column tells us the values of 𝑔 for these same values of 𝑥. For example, 𝑔 of four is equal to eight.
In the question, we’re asked to determine 𝑓 of 𝑔 of two and 𝑔 of 𝑓 of one. So, let’s think about what each of these means. 𝑓 of 𝑔 of 𝑥 is the composite function we get when we take an input, apply the function 𝑔 first, and then apply the function 𝑓 to the result. So, for 𝑓 of 𝑔 of two, we’re taking the input value two, applying 𝑔, and then applying 𝑓 to whatever value we have.
Let’s look at the table. We can see that 𝑔 of two is equal to four. To find 𝑓 of 𝑔 of two, we need to take this value as the input to the function 𝑓. So, 𝑓 of 𝑔 of two is equal to 𝑓 of four. Looking at the table again, this time the second column, we see that 𝑓 of four is equal to 12. So, we found the first answer. 𝑓 of 𝑔 of two is equal to 12.
The second thing we’re asked to evaluate is 𝑔 of 𝑓 of one. Now this time the functions have been composed in the opposite order. We’re applying the function 𝑓 first and then applying the function 𝑔 to the result. Starting with an input value of one then, we see from the table that 𝑓 of one is equal to three. Then, to find 𝑔 of 𝑓 of one, we take this value of three as the input to the function 𝑔. 𝑔 of 𝑓 of one is equal to 𝑔 of three. Returning to the table and looking in the third column, we see that 𝑔 of three is equal to six.
So, we’ve answered both parts of the question. 𝑓 of 𝑔 of two is equal to 12, and 𝑔 of 𝑓 of one is equal to six.
