Video Transcript
Let 𝑓 of 𝑥 equal 𝑥 plus two 𝑏 and the function 𝑔 of 𝑥 equal 𝑏. Find 𝑓 of five plus 𝑔 of negative two given that 𝑓 of negative seven plus 𝑔 of negative three equals negative 34.
First, let’s just write out what we know. 𝑓 of 𝑥 equals 𝑥 plus two 𝑏, and 𝑔 of 𝑥 equals 𝑏. We also know that 𝑓 of negative seven plus 𝑔 of negative three equals negative 34. In order to solve our problem, we’ll need to know what this 𝑏-value is. And since we’ve been given this third equation, we can use this information to solve for 𝑏. To do that, we’ll plug in what we know about 𝑓 of 𝑥 for 𝑓 of negative seven. Since 𝑓 of 𝑥 equals 𝑥 plus two 𝑏, 𝑓 of negative seven will equal negative seven plus two 𝑏. And for 𝑔 of 𝑥, 𝑔 of negative three just equals 𝑏.
Based on the information we were given, we can say then that negative seven plus two 𝑏 plus 𝑏 has to equal negative 34. Our goal is to solve for 𝑏, so we combine like terms. Two 𝑏 plus 𝑏 equals three 𝑏. And then we add seven to both sides of the equation. Negative 34 plus seven equals negative 27. And from there, we’ll divide both sides of the equation by three, which tells us that 𝑏 equals negative nine. We want to take this information and plug it back into our 𝑓 of 𝑥 and 𝑔 of 𝑥 functions. 𝑓 of 𝑥 will then be equal to 𝑥 plus two times negative nine. Two times negative nine is negative 18, which we can write as 𝑥 minus 18. 𝑓 of 𝑥 is equal to 𝑥 minus 18.
And since 𝑔 of 𝑥 is only the constant 𝑏, 𝑔 of 𝑥 is equal to negative nine. But remember, we’re trying to solve 𝑓 of five plus 𝑔 of negative two. To find 𝑓 of five, we substitute the value five in for 𝑥 in the equation so that we have five minus 18. And for 𝑔 of negative two, there’s no 𝑥-variable in our function. The function is equal to negative nine for all 𝑥-values. And that means 𝑔 of negative two is equal to negative nine. To solve for 𝑓 of five plus 𝑔 of negative two, we need to say five minus 18 plus negative nine or five minus 18 minus nine, which equals negative 22.
