Math Problem Statement

f(x) = 2x^2 - 3x + 1, g(x) = f(2x) + 3, find g(x)

Solution

You provided two functions and are tasked with finding g(x)g(x). Here’s the breakdown:

Given:

  1. f(x)=2x23x+1f(x) = 2x^2 - 3x + 1
  2. g(x)=f(2x)+3g(x) = f(2x) + 3

We need to substitute 2x2x into the function f(x)f(x).

Step 1: Find f(2x)f(2x)

Start with the given function f(x)=2x23x+1f(x) = 2x^2 - 3x + 1, and substitute 2x2x in place of xx:

f(2x)=2(2x)23(2x)+1f(2x) = 2(2x)^2 - 3(2x) + 1

Simplifying each term:

f(2x)=2(4x2)6x+1f(2x) = 2(4x^2) - 6x + 1 f(2x)=8x26x+1f(2x) = 8x^2 - 6x + 1

Step 2: Calculate g(x)g(x)

Now, substitute f(2x)f(2x) into the equation for g(x)g(x):

g(x)=f(2x)+3=(8x26x+1)+3g(x) = f(2x) + 3 = (8x^2 - 6x + 1) + 3 g(x)=8x26x+4g(x) = 8x^2 - 6x + 4

Final Answer:

g(x)=8x26x+4g(x) = 8x^2 - 6x + 4

Would you like any further details or clarification?

Here are 5 related questions to deepen your understanding:

  1. How would the result change if g(x)=f(3x)+2g(x) = f(3x) + 2?
  2. Can you find the derivative g(x)g'(x) of the function g(x)g(x)?
  3. What would happen if f(x)f(x) were a cubic function instead of a quadratic one?
  4. How does changing the function from f(2x)f(2x) to f(x/2)f(x/2) affect g(x)g(x)?
  5. Can you find the inverse of g(x)g(x)?

Tip: When substituting variables, be careful with distributing powers and constants to ensure accuracy in each step.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Substitution

Formulas

f(x) = 2x^2 - 3x + 1
g(x) = f(2x) + 3

Theorems

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Suitable Grade Level

Grades 9-11