Question Video: Evaluating Composite Functions at a Given Value

Video Transcript

If 𝑓 of 𝑥 equals three minus 𝑥 squared and 𝑔 of 𝑥 equals two 𝑥 plus four, find 𝑓 of 𝑔 of one.

Now, let’s just be clear on the notation used in the question. 𝑓 of 𝑔 of one means the composite function 𝑓 of 𝑔 of 𝑥 evaluated at 𝑥 equals one. We may also see this written with a small circle between the two letters. Now, what we need to remember is that the composite function 𝑓 of 𝑔 of 𝑥 means the function we get when we apply 𝑔 first and then apply 𝑓 to the result. It doesn’t mean the product of the functions 𝑓 and 𝑔. We’re going to look at two ways of answering this question. In the first method, we’re going to substitute 𝑥 equals one right at the start. So, we’re going to find 𝑔 of one and then evaluate 𝑓 for this value. 𝑔 of 𝑥 is the function two 𝑥 plus four, so 𝑔 of one will be two multiplied by one plus four, which is equal to six.

We’re now going to take this value and evaluate the function 𝑓. So, 𝑓 of 𝑔 of one will become simply 𝑓 of six. 𝑓 of 𝑥 is the function three minus 𝑥 squared, so 𝑓 of six will be three minus six squared. That’s three minus 36, which is equal to negative 33. So, in this method, we evaluated 𝑔 of one, first of all, and then we took this as our input for the second function. Overall, we found that 𝑓 of 𝑔 of one is equal to negative 33. The second approach we could take is to find a general algebraic expression for the composite function 𝑓 of 𝑔 of 𝑥 and then evaluate it when 𝑥 is equal to one. This is probably more complicated in this case, but this method would be useful if we were asked to find 𝑓 of 𝑔 of 𝑥 for multiple different 𝑥-values.

So, let’s see what this looks like. We’re looking to find the general composite function 𝑓 of 𝑔 of 𝑥. Remember, 𝑔 of 𝑥 is the function two 𝑥 plus four. So, replacing 𝑔 of 𝑥 with two 𝑥 plus four, we’re now looking to find the function 𝑓 of two 𝑥 plus four. What we do then is we take the expression two 𝑥 plus four as our input to the function 𝑓. 𝑓 of 𝑥 is the function three minus 𝑥 squared. So, 𝑓 of two 𝑥 plus four is the function three minus two 𝑥 plus four all squared. We can keep our composite function in this form or we can distribute the parentheses and simplify, if we wish. And it will give 𝑓 of 𝑔 of 𝑥 equals negative four 𝑥 squared minus 16𝑥 minus 13.

We now have a general expression for 𝑓 of 𝑔 of 𝑥. But remember, we were asked to evaluate this when 𝑥 equals one. So, the final step is to substitute 𝑥 equals one. This gives negative four minus 16 minus 13, which is equal to negative 33, the same answer as we found using our previous method. So in both cases, we found that 𝑓 of 𝑔 of one is equal to negative 33. The first method is probably more straightforward for this particular question. But if we needed to evaluate 𝑓 of 𝑔 of 𝑥 for multiple 𝑥-values, then the second method may well be more efficient.