Given that 𝑓 of 𝑥 equals the fifth root of 𝑥 and 𝑔 of 𝑥 equals 𝑥 plus 46 to the fifth power, find and simplify an expression for 𝑓 of 𝑔 of 𝑥.
Let’s begin by recalling what we mean by this notation, which I read as 𝑓 of 𝑔 of 𝑥. This means that we take an input value 𝑥, apply the function 𝑔 to it, and then we apply the function 𝑓 to the result. This is called a composite function. We’re applying one function and then another.
We want to find and simplify an algebraic expression for what this composite function will be. So let’s see what this looks like. We begin by applying the function 𝑔. 𝑔 of 𝑥 is the function that takes an input value, adds 46, and then raises it to the fifth power. So we have 𝑔 of 𝑥 equals 𝑥 plus 46 to the fifth power. Next, we’re going to apply the function 𝑓, which means we take this expression for 𝑔 of 𝑥 as our input value to the function 𝑓. So, 𝑓 of 𝑔 of 𝑥 is equal to 𝑓 of 𝑥 plus 46 to the fifth power. Now, 𝑓 is the function that takes the fifth root of its input. So, 𝑓 of 𝑥 plus 46 to the fifth power is the fifth root of 𝑥 plus 46 to the fifth power.
We’ve now found an expression for 𝑓 of 𝑔 of 𝑥, but we need to simplify it. Well, we know that raising a value or expression to the fifth power and then taking the fifth root are inverse operations. So, if we do this to the expression 𝑥 plus 46, we just get back to the expression 𝑥 plus 46 again. We can demonstrate this a little more rigorously using laws of indices. An 𝑛th root defines a reciprocal exponent. The 𝑛th root of 𝑥 is equal to 𝑥 to the power of one over 𝑛. So a fifth root corresponds to a power of one-fifth. We therefore have 𝑥 plus 46 to the power of five to the power of one-fifth.
We can then recall that if 𝑛 is odd, 𝑥 to the power of 𝑛 to the power of one over 𝑛 is equal to 𝑥, which we find by multiplying the indices together. If 𝑛 is even, we have to be a little bit more careful. And we have that 𝑥 to the 𝑛th power to the power of one over 𝑛 is equal to the absolute value of 𝑥. But as we’re working with an odd value of 𝑛 here, 𝑛 is equal to five, this doesn’t apply.
So we’ve found that a simplified expression for the composite function 𝑓 of 𝑔 of 𝑥 is 𝑥 plus 46.