Determine the values of 𝑥 for which 𝑓 of 𝑔 of 𝑥 equals 407, where 𝑓 of 𝑥 equals 𝑥 squared plus 83 and 𝑔 of 𝑥 equals three 𝑥.
We begin by recalling that this notation 𝑓 and then a little circle and then 𝑔 means the function 𝑓 of 𝑔 of 𝑥. This means that we take an input value 𝑥, we apply the function 𝑔, and then we apply the function 𝑓 to the result. In other words, we take the function 𝑔 of 𝑥 as the input to the function 𝑓.
Now, we’re given that 𝑓 of 𝑔 of 𝑥 is equal to 407 for certain values of 𝑥 and asked to determine what those values are. We’ll begin by finding an algebraic expression for the function 𝑓 of 𝑔 of 𝑥. So we start with an input value 𝑥. And then 𝑔 of 𝑥 is the function that multiplies this input by three. 𝑔 of 𝑥 is equal to three 𝑥. We’re then going to apply the function 𝑓 to the result. So 𝑓 of 𝑔 of 𝑥 is 𝑓 of three 𝑥. We take the expression for 𝑔 of 𝑥 and we use it as our input to the function 𝑓.
Now, looking at the function 𝑓 of 𝑥, this is the function that takes its input value, squares it, and then adds 83. So we take the input value or the input expression of three 𝑥, square it, and add 83. Now, we must be really careful here and make sure that we square both the 𝑥 and the three. So the expanded version of 𝑓 of three 𝑥 is nine 𝑥 squared plus 83, not three 𝑥 squared plus 83. So we found our algebraic expression for the composite function 𝑓 of 𝑔 of 𝑥.
Next, we need to set this expression equal to 407 and then solve the equation. We have nine 𝑥 squared plus 83 equals 407. We can subtract 83 from each side of the equation to give nine 𝑥 squared equals 324. And then we divide by nine, giving 𝑥 squared is equal to 36. To find the value or values of 𝑥, we need to take the square root of each side of this equation. We mustn’t forget to take plus or minus the square root. So we have 𝑥 is equal to plus or minus the square root of 36, which is positive or negative six. We found then that there are two values of 𝑥 for which 𝑓 of 𝑔 of 𝑥 is equal to 407: positive and negative six.
We can check our answers by substituting these values into the function 𝑔 and then substituting the output into the function 𝑓. For example, when 𝑥 is equal to six, 𝑔 of 𝑥, 𝑔 of six, is three times six, which is 18. Then, 𝑓 of 𝑔 of six is 𝑓 of 18. That’s 18 squared plus 83, or 324 plus 83, which is indeed equal to 407. For 𝑥 equals negative six, 𝑔 of negative six will be negative 18. So 𝑓 of 𝑔 of negative six will be negative 18 squared plus 83. But of course negative 18 squared is the same as 18 squared. So we know that this will also give us 407.
We’ve determined that there are two values of 𝑥 for which the composite function 𝑓 of 𝑔 of 𝑥 is equal to 407. And those two values are six and negative six.