Missing piece Let g(x) = x² + 3

Question

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.

(g o ƒ ) (x) = x⁴ + 3

Accepted Answer

Hello. Today we're going to be determining the function u of x using the following functions. So we are told that we have the composite function v of u of x, and this composite function is equal to x raised to the power of 4+7. We are also told that the function v of x is equal to x squared+7. So how can we use these two functions in order to find the function u of x? Well, if we take a look at the given composite function, we can rewrite the composite function as v of u of x. In order to get this composite function, we need to plug in the function u into the v function. Now again, we do know that v is equal to x squared plus 7. If we were to plug in the u of x function into this v of x function, we can write the composite function as u of x squared plus 7. Now one thing to note is that one similarity between the composite function and the v of x function is that both of them share the same constant positive 7. That means that when we plug in the function u of x into v of x, the constant of positive 7 remains the same. However, the exponent of the x term does change. That means u of x has to be some type of exponential value. So for now let's go ahead and state that u is equal to x raised to the power of n. This is going to be the general form of the u function and that means we need to solve for the variable n. So, if we were to substitute x to the power of n with u of x we can rewrite the composite function as v of u of x equal to x to the power of n raised to the power of 2 plus 7. Now on the right hand side of this equation, x raised to the power of n raised to the power of 2 can simplify as x to the power of 2 n+7. Now on the left hand side, we have the composite function v of u of x. We do know that the composite function v of u of x is equal to x raised to the power of 4+7. We can substitute the composite function as x raised to the power of 4+7 on the left hand side of the equation. So the equation we are left with is x raised to the power of 4+7equal to x raised to the power of 2 n+7. Now what we want to do is we want to simplify the equation in order to solve for n. The first thing we can do is we can eliminate the constants of 7 by subtracting 7 to both sides of the equation. Doing this will allow us to simplify the equation to x raised to the power of 4 equal to x raised to the power of 2 n. Now one thing to note is that on both sides of the equation, we have the same exponential value with the base of x. What this means is that we can take the exponents and set them equal to each other. So that will allow us to rewrite the equation as 4 is equal to 2 n. And in order to solve for n, we'll just need to divide both sides of the equation by 2 and that will leave us with n is equal to 2. If we plug in this value of n into the general form of the u of x function, we can state that u of x is equal to x raised to the power of 2. This is going to be the u of x function that we are looking for. And with that being said, the answer to this problem is going to be c. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Missing piece Let g(x) = x² + 3 Find a function ƒ that produces the given composition.

(g o ƒ ) (x) = x⁴ + 3

Key Concepts

Function Composition

Quadratic and Polynomial Functions

Finding Inverse Relationships

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