Math Problem Statement

Solution

Let's calculate f(g(x))f(g(x)), which involves substituting g(x)=2x3+xg(x) = 2x^3 + x into the function f(x)=2x44x+2f(x) = 2x^4 - 4x + 2.

Given:

  • f(x)=2x44x+2f(x) = 2x^4 - 4x + 2
  • g(x)=2x3+xg(x) = 2x^3 + x

To find f(g(x))f(g(x)), substitute g(x)g(x) into f(x)f(x):

f(g(x))=2(g(x))44(g(x))+2f(g(x)) = 2(g(x))^4 - 4(g(x)) + 2

Substituting g(x)=2x3+xg(x) = 2x^3 + x into the equation:

f(g(x))=2(2x3+x)44(2x3+x)+2f(g(x)) = 2(2x^3 + x)^4 - 4(2x^3 + x) + 2

To expand (2x3+x)4(2x^3 + x)^4, we can use the binomial expansion or simplify for a particular value of xx. However, expanding the expression exactly would be complex, so unless a specific xx value is given, we'll keep it in its expanded polynomial form.

Simplifying the linear term:

4(2x3+x)=8x34x-4(2x^3 + x) = -8x^3 - 4x

Thus, the function becomes:

f(g(x))=2(2x3+x)48x34x+2f(g(x)) = 2(2x^3 + x)^4 - 8x^3 - 4x + 2

If you need the full expansion or evaluation for a specific value of xx, let me know!

Would you like more details or have any questions about this calculation?

Here are some related questions:

  1. What is the derivative of f(g(x))f(g(x))?
  2. Can you evaluate f(g(1))f(g(1))?
  3. How would you describe the composition of functions?
  4. What happens when you compose g(f(x))g(f(x))?
  5. How does the order of function composition affect the result?

Tip: When working with function compositions, always substitute the inner function correctly into the outer function before simplifying or expanding.

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Math Problem Analysis

Mathematical Concepts

Function composition
Polynomial functions
Binomial expansion

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12