Final answer:
The correct composite functions presented in the question are g(h(x)) = 3x⁴⁸ and ƒ(h(x)) = 2x²⁴. The others, ƒ(g(x)) and h(ƒ(x)), are incorrectly stated.
Step-by-step explanation:
The student is asking about composite functions and whether the given expressions are correct. For composite functions, you substitute one function into another. The rule for composing functions is to take the output from the first function (the inner function) and use it as the input for the second function (the outer function). When dealing with functions that have exponents, you will multiply the exponents when the functions are composed. Let's evaluate each composite function one by one.
f(g(x)):
Given ƒ(x) = 2x⁴, and g(x) = 3x⁸, we want to find ƒ(g(x)). Substituting g(x) into ƒ(x), we have ƒ(3x⁸) = 2(3x⁸)⁴ = 2·(3⁴)·(x⁸⁴) = 2·(81)·x³² = 162x³², which is not equal to 81x³². Therefore, ƒ(g(x)) = 81x³² is incorrect.
h(f(x)):
Given h(x) = x⁶ and ƒ(x), we find h(ƒ(x)). Substituting ƒ(x) into h(x), we get h(2x⁴) = (2x⁴)⁶ = 2⁶x²⁴ = 64x²⁴. Thus, h(f(x)) = 64x±⁰ is incorrect.
g(h(x)):
Given g(x) = 3x⁸ and h(x), we find g(h(x)). Substituting h(x) into g(x), we get g(x⁶) = 3( x⁶ )⁸ = 3x´⁸ = 3x´⁸. Thus, g(h(x)) = 3x±⁴ is correct.
f(h(x))
Applying the same method, we find ƒ(h(x)) by substituting h(x) into ƒ(x), resulting in ƒ(x⁶) = 2( x⁶ )⁴ = 2x²⁴ which means that ƒ(h(x)) = 2x²⁴ is correct.