If ƒ(x) = √x and g(x) = x³-2 and , simplify the expre...

Question

If ƒ(x) = √x and g(x) = x³-2 and , simplify the expressions (ƒ o g) (3), (ƒ o ƒ) (64), (g o ƒ) (x) and (ƒ o g) (x)

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the values of u of v of x and v of u of x for u of x equals x to the 7th power and v of x equals x to the 4th power minus 6. For our answer choices, a says that you are v of x equals x to the 4th power minus 6. All raised to the 7th power and v of u of x equals x to the 28th power minus 6. B says u of v of x equals x to the 4th power minus 6 all raised to the 7th power and v of u of x equals x to the 11th power minus 6. C says u of v of x equals x to the 20th power minus 6 to the 7th power And v of u of x equals x to the 20th power minus 6. And d says that u of v of x equals x to the 20th power minus 6. And v of u of x equals x to the 4th power minus 6 all raised to the 7th power. Now, what are we trying to figure out here? Well, both of these are composite functions and recall that a composite to find a composite function, it means to put one function into another. So when we see let's take you off the of x. Okay? So u of v of x. What it really means is that we're putting the function v of x into the function u. Now, notice that u of x is x to the 7th power. But since we're putting v of x into the function, wherever we see x, we're going to put v of x. And recall that v of x is x to the 4th power minus 6. So our function would be x to the 4th power minus 6 all raised to the 7th power. How about v of u of x? Well, v of u of x, like we said earlier, means that no, we're putting u into v. So it's going to be v of u of x and that means we're going to look at v of x and anywhere we see our input x, we're going to put the function u of x. So instead of x to the 4th, no, it's going to be x to the 7th power raised to the 4th and we subtract 6. X to the 7th power raised to the 4th power, recall that for the laws of indices, it says that if we are raising an index or raising a power to another power, then we multiply both those powers. So what that means here is that if we're raising x to the 7th to the 4th power, we're going to multiply 7 by 4 and 7 multiplied by 4 equals 28. So v of u of x equals x to the 28th power minus 6 and you have v of x equals x to the 4th power minus 6 all raised to the 7th power. When we look back on our answer choices, that would have been answer choice a. Thanks a lot for watching everyone. I hope this video helped.

If ƒ(x) = √x and g(x) = x³-2 and , simplify the expressions (ƒ o g) (3), (ƒ o ƒ) (64), (g o ƒ) (x) and (ƒ o g) (x)

Key Concepts

Function Composition

Square Root Function

Polynomial Functions

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