Math Problem Statement

(h ∘ h)(u) = ? where f(u) = u + 2 and h(u) = u^2 - 3

Solution

The functions given are:

f(u)=u+2f(u) = u + 2 h(u)=u23h(u) = u^2 - 3

The problem asks to find (hh)(u)(h \circ h)(u), which is the composition of hh with itself, meaning h(h(u))h(h(u)).

Step-by-step solution:

  1. Start by computing h(u)h(u): h(u)=u23h(u) = u^2 - 3

  2. Now, substitute h(u)h(u) into hh itself: h(h(u))=h(u23)h(h(u)) = h(u^2 - 3)

  3. Use the expression of h(x)=x23h(x) = x^2 - 3, so we substitute u23u^2 - 3 into this formula: h(u23)=(u23)23h(u^2 - 3) = (u^2 - 3)^2 - 3

  4. Expand (u23)2(u^2 - 3)^2: (u23)2=u46u2+9(u^2 - 3)^2 = u^4 - 6u^2 + 9

  5. Now substitute this back: h(h(u))=(u46u2+9)3h(h(u)) = (u^4 - 6u^2 + 9) - 3

  6. Simplify: h(h(u))=u46u2+6h(h(u)) = u^4 - 6u^2 + 6

Thus, (hh)(u)=u46u2+6(h \circ h)(u) = u^4 - 6u^2 + 6.

Would you like any further details or have questions? Here are some related questions for deeper understanding:

  1. How would you compute (fh)(u)(f \circ h)(u)?
  2. What happens if you reverse the composition to find (hf)(u)(h \circ f)(u)?
  3. Can you explore the domain and range of h(u)h(u) and f(u)f(u)?
  4. What is the behavior of h(h(u))h(h(u)) for large values of uu?
  5. How can you find the inverse of f(u)f(u) or h(u)h(u) if they exist?

Tip: When dealing with function composition, always substitute the inner function into the outer function carefully, keeping track of all terms!

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Composition
Polynomial Functions

Formulas

(h ∘ h)(u) = h(h(u))
(u^2 - 3)^2 = u^4 - 6u^2 + 9

Theorems

Function Composition

Suitable Grade Level

Grades 10-12