Find functions ƒand g such that ƒ(g(x))...

Question

Find functions ƒand g such that ƒ(g(x)) = (x² +1)⁵ . Find a different pair of functions ƒ and g that also satisfy ƒ(g(x)) = (x² +1)⁵

Accepted Answer

Welcome back, everyone. In this problem, we want to figure out which of the following pair of u of x and v of x values is correct for u of v of x equals x to the 3rd power minus 5 all raised to the 4th power. For our answer choices, a says that you have x equals x cubed minus 5 and v of x equals x to the 4th power. B says you have x equals x to the 4th power and v of x equals x cubed minus 5. C says you have x equals x to the 4th power minus 5 and v of x equals x cubed. And these says you have x equals x cubed and v of x equals x to the 4th power minus 5. Now what do we know here? Well, we're looking for the pair of u of x and v of x functions that gives us the correct composite function. You are V of X. And recall that for our composite function, it means that we put one function into another. So to figure out which pair is correct for you are V of X, we need to put v of x because that's what you are v of x means. You are v of x means putting v of x into u of x. So what we need to do is from our answer choices, figure out for which functions when we put the V of X into U of X, we get X Q minus 5 or raised to the 4th power. So let's go ahead and try our values. Let's let's start with a. So far, a where you have x equals Let me put it in blue actually. And for a, where you have x equals x cubed minus 5 and v of x equals x to the 4th power, then that means you have v of x means you're putting v of x that is x to the 4th into our function x cubed minus 5. So that means it's going to become instead of x cubed, it's going to be x to the 4th power cubed minus 5. Now x to the 4th power cubed. Another thing we can recall is that when we're raising a power to another power, so let's say we have a to the power of b raised to the power of c, then we multiply it to the powers. So that would be a to the power of b multiplied by c. Likewise here, that means we're going to multiply 4 by 3 for our powers and 4 multiplied by 3 equals 12. So for a, u of z of x equals x to the 12th power minus 5. But that's not our function. So that means it could not be correct. Let's try to do it with b. So for b, u of v of x means that we're going to put v of x which is x cubed minus 5 into u of x which is x to the 4th power. So that means we're going to replace x with x cubed minus 5 and then we're going to write it to the 4th power. Now, is the function in b the same that we have in our problem? Yes. That's correct. Therefore, that's how we know that our correct answer is b. If you wanted to, we could try c and d just to make sure that we are right. So when we look at answer choice C, u of v of x means that we'd be putting x cubed into x to the 4th minus 5. So that would be x cubed to the 4th power minus 5, which is x to the 12th power minus 5. So we know for sure that c isn't correct. And for d, u of v of x means that we're putting v of x, which is x to the 4th to x to the 4th minus 5 into x cubed. So that would be x to the 4th minus 5 all cubed. And again, that one is not our function. So that's how we know for sure that the pair of u of x and v of x functions would be correct in answer choice b. Thanks a lot for watching everyone. I hope this video helped.

Find functions ƒand g such that ƒ(g(x)) = (x² +1)⁵ . Find a different pair of functions ƒ and g that also satisfy ƒ(g(x)) = (x² +1)⁵

Key Concepts

Composition of Functions

Polynomial Functions

Function Decomposition

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