Question Video: Forming and Simplifying Composite Functions Involving Radical and Polynomial Functions | Nagwa Question Video: Forming and Simplifying Composite Functions Involving Radical and Polynomial Functions | Nagwa

# Question Video: Forming and Simplifying Composite Functions Involving Radical and Polynomial Functions Mathematics

Given that π(π₯) = the fifth root of π₯ and π(π₯) = (π₯ + 46)β΅, find and simplify an expression for (π β π)(π₯).

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### Video Transcript

Given that π of π₯ equals the fifth root of π₯ and π of π₯ equals π₯ plus 46 to the fifth power, find and simplify an expression for π of π of π₯.

Letβs begin by recalling what we mean by this notation, which I read as π of π of π₯. This means that we take an input value π₯, apply the function π to it, and then we apply the function π to the result. This is called a composite function. Weβre applying one function and then another.

We want to find and simplify an algebraic expression for what this composite function will be. So letβs see what this looks like. We begin by applying the function π. π of π₯ is the function that takes an input value, adds 46, and then raises it to the fifth power. So we have π of π₯ equals π₯ plus 46 to the fifth power. Next, weβre going to apply the function π, which means we take this expression for π of π₯ as our input value to the function π. So, π of π of π₯ is equal to π of π₯ plus 46 to the fifth power. Now, π is the function that takes the fifth root of its input. So, π of π₯ plus 46 to the fifth power is the fifth root of π₯ plus 46 to the fifth power.

Weβve now found an expression for π of π of π₯, 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 π₯ plus 46, we just get back to the expression π₯ plus 46 again. We can demonstrate this a little more rigorously using laws of indices. An πth root defines a reciprocal exponent. The πth root of π₯ is equal to π₯ to the power of one over π. So a fifth root corresponds to a power of one-fifth. We therefore have π₯ plus 46 to the power of five to the power of one-fifth.

We can then recall that if π is odd, π₯ to the power of π to the power of one over π is equal to π₯, which we find by multiplying the indices together. If π is even, we have to be a little bit more careful. And we have that π₯ to the πth power to the power of one over π is equal to the absolute value of π₯. But as weβre working with an odd value of π here, π is equal to five, this doesnβt apply.

So weβve found that a simplified expression for the composite function π of π of π₯ is π₯ plus 46.