Math Problem Statement

Refer to the functions f, g, and h and find the given function (f ∘ g ∘ h)(x). The functions are: f(x) = 3x + 7, g(x) = x^2, and h(x) = ∛x.

Solution

You are tasked with finding the composite function (fgh)(x)(f \circ g \circ h)(x) using the given functions:

  • f(x)=3x+7f(x) = 3x + 7
  • g(x)=x2g(x) = x^2
  • h(x)=x3h(x) = \sqrt[3]{x}

The composition of functions (fgh)(x)(f \circ g \circ h)(x) means we need to evaluate the functions in the following order:

  1. First, find h(x)h(x), which is h(x)=x3h(x) = \sqrt[3]{x}.
  2. Then, substitute h(x)h(x) into g(x)g(x), so compute g(h(x))=g(x3)=(x3)2=x2/3g(h(x)) = g(\sqrt[3]{x}) = (\sqrt[3]{x})^2 = x^{2/3}.
  3. Finally, substitute g(h(x))g(h(x)) into f(x)f(x), so compute f(g(h(x)))=f(x2/3)=3(x2/3)+7f(g(h(x))) = f(x^{2/3}) = 3(x^{2/3}) + 7.

Thus, the final composite function is: (fgh)(x)=3x2/3+7(f \circ g \circ h)(x) = 3x^{2/3} + 7

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you evaluate the composition of more than two functions?
  2. What are the properties of cube roots in function composition?
  3. How does exponentiation work when composing functions with fractional exponents?
  4. Can you explain how function composition changes if one of the functions is not one-to-one?
  5. What happens if we reverse the order of function composition in this case?

Tip: When composing functions, always start from the innermost function and work outward step by step.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Exponents
Cube Roots

Formulas

f(x) = 3x + 7
g(x) = x^2
h(x) = ∛x
Composite function (f ∘ g ∘ h)(x)

Theorems

Function Composition Theorem

Suitable Grade Level

Grades 9-12