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

### Given:
1. \( f(x) = 2x^2 - 3x + 1 \)
2. \( g(x) = f(2x) + 3 \)

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

### Step 1: Find \( f(2x) \)
Start with the given function \( f(x) = 2x^2 - 3x + 1 \), and substitute \( 2x \) in place of \( x \):

\[
f(2x) = 2(2x)^2 - 3(2x) + 1
\]

Simplifying each term:

\[
f(2x) = 2(4x^2) - 6x + 1
\]
\[
f(2x) = 8x^2 - 6x + 1
\]

### Step 2: Calculate \( g(x) \)
Now, substitute \( f(2x) \) into the equation for \( g(x) \):

\[
g(x) = f(2x) + 3 = (8x^2 - 6x + 1) + 3
\]
\[
g(x) = 8x^2 - 6x + 4
\]

### Final Answer:
\[
g(x) = 8x^2 - 6x + 4
\]

Would you like any further details or clarification? 

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

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

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

Learn how to solve the function g(x) = f(2x) + 3 for f(x) = 2x^2 - 3x + 1 using substitution. This step-by-step solution covers key algebraic concepts and function manipulation. Suitable for students in Grades 9-11.

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