Video Transcript
The following tables are partially filled for functions 𝑓 and 𝑔 that are inverses of each other. Determine the values of 𝑎, 𝑏, 𝑐, 𝑑, and 𝑒.
We are told that the functions 𝑓 and 𝑔 are inverses of each other. This means that the input of one function will be the output of the other. The function 𝑓 of 𝑥 contains six ordered pairs. They are one, negative 31; two, 𝑎; three, negative 19; four, negative 10; 𝑑, one; and six, 14. The function 𝑔 of 𝑥 also contains six ordered pairs. They are negative 31, one; negative 26, two; negative 19, 𝑏; negative 10, 𝑐; one, five; and 𝑒, six.
Looking at the first set of pairs, we notice that an input of one gives an output of negative 31 for 𝑓 of 𝑥 and an input of negative 31 gives an output of one for 𝑔 of 𝑥. We know that this behavior should hold as 𝑓 and 𝑔 are inverses. This means that 𝑎 must be equal to negative 26. When 𝑥 equals negative 26, 𝑔 of 𝑥 equals two. Therefore, when 𝑥 equals two, 𝑓 of 𝑥 must equal negative 26. 𝑏 must be equal to three as when 𝑥 equals three, 𝑓 of 𝑥 equals negative 19. And when 𝑥 equals negative 19, 𝑔 of 𝑥 must equal three.
Using the same pattern, we see that 𝑐 is equal to four. 𝑑 must be equal to five as when 𝑥 equals one, 𝑔 of 𝑥 is equal to five. Therefore, when 𝑥 is five, 𝑓 of 𝑥 must equal one. Finally, we see that 𝑒 is equal to 14. When 𝑥 equals six, 𝑓 of 𝑥 equals 14. And when 𝑥 equals 14, 𝑔 of 𝑥 equals six. The values of 𝑎 to 𝑒 are negative 26, three, four, five, and 14, respectively.
