Missing piece Let g(x) = x² + 3

Question

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

(ƒ o g) (x) = x⁴ + 6x² + 9

Accepted Answer

Hello. Today we're going to be determining the u of x function for the following functions. So we are given a composite function. We are told that u of v of x is equal to 2x to the 4th power plus 8x squared plus 8. We are also told that the function v of x is equal to x squared plus 2. So how can we determine the function u of x from these two functions? Well, we can rewrite the notation for the composite function as u of v of x. What this means is that the function v of x is the input value for the function u. So if we want to determine the function u, we can start by simplifying the composite function. Now notice that each coefficient in the composite function is a factor of 2. We can start by factoring out a 2 from the composite function and that will allow us to rewrite the function as 2 multiplied by x to the 4th plus 4x squared plus 4. Next, if we want to simplify this further, we're going to need to use a substitution. We're going to allow y to equal to x squared. And if we apply this substitution, we can rewrite the composite function as 2 multiplied by y squared plus 4 y plus 4. Y squared+4y+4 is in the form of a factorable trinomial and if we factor this trinomial we can rewrite the expression or we can rewrite the composite function as 2 multiplied by the quantity (y+2) multiplied by (y+2). Next, notice that we have 2 quantities of y+2. We can combine them into a single quantity and rewrite the function as 2 multiplied by the quantity (y+2) squared Now we're going to resubstitute in the value of y for x squared and we can rewrite the function as 2 multiplied by the quantity (x squared plus 2) squared. Now notice that we have the quantity x squared plus 2. This is the same function as our v of x function. So we can replace x squared plus 2 with v of x and what this means is that the composite function is given to us as 2 multiplied by the quantity v of x raised to the power of 2. Now, what this means is that v of x is acting as the input value for the function u of x. So if we replace v of x with just x that means that the function u of x is equal to 2 x squared, and this is going to be the value or the function u. And with that being said, the answer to this problem is going to be d. 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.

(ƒ o g) (x) = x⁴ + 6x² + 9

Key Concepts

Function Composition

Quadratic Functions

Polynomial Degree

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