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Given 𝑓(𝑥) = 3𝑥 − 1 and 𝑔(𝑥) = 𝑥² + 1, which of the following expressions gives (𝑓 ∘ 𝑔)(𝑥)? [A] 3𝑥² + 3 [B] 9𝑥² − 6𝑥 + 3 [C] 3𝑥² +2 [D] 3𝑥² [E] 9𝑥² − 6𝑥 + 2
Given 𝑓 of 𝑥 equals three 𝑥 minus one and 𝑔 of 𝑥 equals 𝑥 squared plus one, which of the following expressions gives 𝑓 of 𝑔 of 𝑥? Is it option A) three 𝑥 squared plus three, option B) nine 𝑥 squared minus six 𝑥 plus three, option C) three 𝑥 squared plus two, option D) three 𝑥 squared, or option E) nine 𝑥 squared minus six 𝑥 plus two.
So what is the thing here which is represented by this 𝑓 and then this small circle and then this 𝑔? Well, it’s a new function which you get by composing the functions 𝑓 and 𝑔. We can explain what this function does by showing its effect on an arbitrary input 𝑥. We have that 𝑓 of 𝑥 is equal to three 𝑥 minus one and 𝑔 of 𝑥 is equal to 𝑥 squared plus one and we’d like to have a similar expression on the right-hand side to show what happens to 𝑥 when 𝑓 of 𝑔 is applied instead of 𝑓 or 𝑔 alone.
How do we do this? Well, we use a definition of function composition which tells us that 𝑓 composed with 𝑔, or 𝑓 of 𝑔, of 𝑥 is equal to 𝑓 of 𝑔 of 𝑥. Before we try to find an expression for 𝑓 of 𝑔 of 𝑥, let’s just see where 𝑓 of 𝑔 takes the number two. And of course, we just replace 𝑥 by two wherever we see it in the above definition. So we get that 𝑓 of 𝑔 of two is equal to 𝑓 of 𝑔 of two. We want to evaluate the right-hand side, and we do that by first finding 𝑔 of two and substituting that it.
What is 𝑔 of two? Well, we can find that by using our expression for 𝑔 of 𝑥. 𝑔 of 𝑥 is 𝑥 squared plus one, so 𝑔 of two is two squared plus one which is five. And we substitute that in. So now we see that 𝑓 of 𝑔 of two is 𝑓 of five. Now I need to find 𝑓 of five. How do we do that? We use the expression that we have for 𝑓 of 𝑥. 𝑓 of 𝑥 is three 𝑥 minus one and so 𝑓 of five is three times five minus one which is 14. So when the function 𝑓 of 𝑔 is given the input two, we get the output 14.
How do you find 𝑓 of 𝑔 of 𝑥 though? It’s going to be very similar to finding 𝑓 of 𝑔 of two except we’re going to be dealing with algebraic expressions instead of numbers. So first again, we evaluated the right-hand side, and we do that by first finding 𝑔 of 𝑥. Well, we have an expression for 𝑔 of 𝑥 in the question, so let’s just substitute that expression in. So we get 𝑓 of 𝑥 squared plus one.
Okay. Now what do we do with that? We use the definition for 𝑓, so 𝑓 of 𝑥 is equal to three 𝑥 minus one. What is 𝑓 of 𝑥 squared plus one then? Well, 𝑓 takes an input and returns three times that input minus one. So the answer is: three times 𝑥 squared plus one minus one. And we can simplify this by first applying the distributive property to multiply out the brackets and then collecting the constant terms to get three 𝑥 squared plus two.
Okay. So now that we have our answer, let’s go through the options, and we can see that C, three 𝑥 squared plus two, is our correct answer.
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