Video Transcript
If 𝑓 of 𝑥 plus 𝑔 of 𝑥 equals negative two 𝑥 squared plus six 𝑥 plus three and 𝑓 of 𝑥 equals two 𝑥 squared minus four 𝑥 minus eight, which of the following represents the function 𝑔? A) 𝑔 of 𝑥 equals two 𝑥 minus five. B) 𝑔 of 𝑥 equals negative two 𝑥 plus five. C) 𝑔 of 𝑥 equals negative four 𝑥 squared plus 10𝑥 plus 11. Or D) 𝑔 of 𝑥 equals four 𝑥 squared minus 10𝑥 minus 11.
If we start with 𝑓 of 𝑥, we have two 𝑥 squared minus four 𝑥 minus eight. And we add some 𝑔 of 𝑥 to it. The result is 𝑓 of 𝑥 plus 𝑔 of 𝑥, negative two 𝑥 squared plus six 𝑥 plus three. We can fill in these 𝑔 of 𝑥 boxes with some variables 𝑎, 𝑏, and 𝑐. And then, we can say two 𝑥 squared plus 𝑎 has to equal negative two 𝑥 squared. Negative four 𝑥 plus 𝑏 equals six 𝑥. And negative eight plus 𝑐 equals three. At this point, we can solve each of these equations for the variables. To get 𝑎 by itself, we subtract two 𝑥 squared from both sides. Negative two 𝑥 squared minus two 𝑥 squared equals negative four 𝑥 squared. So we plug negative four 𝑥 squared in for 𝑎.
Now we solve for 𝑏. To do that, we add four 𝑥 to both sides of the equation. 𝑏 equals 10𝑥. And we plug that back into our 𝑔 of 𝑥 equation. Finally solve for 𝑐 by adding eight to both sides. Three plus eight equals 11. 𝑐 equals 11. And we plug that in. This tells us that 𝑔 of 𝑥 equals negative four 𝑥 squared plus 10𝑥 plus 11. If we wanted to double-check, we could say is two 𝑥 squared minus four 𝑥 squared equal to negative two 𝑥 squared? Is negative four 𝑥 plus 10𝑥 equal to six 𝑥? And is negative eight plus 11 equal to three? Only one of our answer choices has 𝑔 of 𝑥 equal to negative four 𝑥 squared plus 10𝑥 plus 11. That’s option C.
We’ve looked at a way to solve this problem. However, we probably could’ve solved it by elimination. In the beginning, we see that 𝑓 of 𝑥 plus 𝑔 of 𝑥 has a term of three. And 𝑓 of 𝑥 has a term of negative eight. We could’ve asked how do we go from negative eight to three. To go from negative eight to three, you need to add 11. And this 11 is positive. Only one of our answer choices has a positive 11 term for 𝑔 of 𝑥. And thus, through elimination, we could’ve chosen option C.
