Math Problem Statement

Find functions f(x) and g(x) so the given function can be expressed as h(x) = f(g(x)). Use non-identity functions for f(x) and g(x). h(x) = (x + 2)^2.

Solution

To express the given function h(x)=(x+2)2h(x) = (x + 2)^2 in the form h(x)=f(g(x))h(x) = f(g(x)), we need to find appropriate functions f(x)f(x) and g(x)g(x).

Step 1: Identify the Inner Function g(x)g(x)

Since the function inside the square is x+2x + 2, we can choose: g(x)=x+2g(x) = x + 2

Step 2: Identify the Outer Function f(x)f(x)

Now, h(x)h(x) is obtained by squaring g(x)g(x). So, the outer function f(x)f(x) should be the squaring function: f(x)=x2f(x) = x^2

Step 3: Express h(x)h(x) in the Form f(g(x))f(g(x))

Substituting g(x)=x+2g(x) = x + 2 and f(x)=x2f(x) = x^2 into the form f(g(x))f(g(x)), we have: h(x)=f(g(x))=(g(x))2=(x+2)2h(x) = f(g(x)) = (g(x))^2 = (x + 2)^2

Thus, the functions are: {f(x),g(x)}={x2,x+2}\{f(x), g(x)\} = \{x^2, x + 2\}

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


Here are 5 related questions you might find interesting:

  1. How would the functions f(x)f(x) and g(x)g(x) change if h(x)=(x3)3h(x) = (x - 3)^3?
  2. Can you express h(x)=sin(x+1)h(x) = \sin(x + 1) in the form h(x)=f(g(x))h(x) = f(g(x))?
  3. What are the functions f(x)f(x) and g(x)g(x) if h(x)=x4h(x) = \sqrt{x - 4}?
  4. How can you express h(x)=e2xh(x) = e^{2x} in the form f(g(x))f(g(x))?
  5. How would you express a polynomial like h(x)=(2x+5)4h(x) = (2x + 5)^4 using f(x)f(x) and g(x)g(x)?

Tip: When breaking down a composite function into f(g(x))f(g(x)), always start by identifying the innermost transformation (which becomes g(x)g(x)) and then determine the outer function f(x)f(x) accordingly.

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Composition
Quadratic Functions

Formulas

h(x) = (x + 2)^2
f(g(x)) = f(x^2)
g(x) = x + 2

Theorems

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Suitable Grade Level

Grades 9-12