given f(z) = 2z^2 + 4, evaluate and fully expand: f( 2x + 1 )

We are given the function:

\[
f(z) = 2z^2 + 4
\]

Now we need to evaluate \( f(2x + 1) \). To do this, we substitute \( 2x + 1 \) for \( z \) in the function:

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

### Step 1: Expand \( (2x + 1)^2 \)
We start by expanding the square:

\[
(2x + 1)^2 = (2x + 1)(2x + 1) = 4x^2 + 4x + 1
\]

### Step 2: Multiply by 2
Next, we multiply the expanded form by 2:

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

### Step 3: Add the constant term
Now, we add the constant 4 from the original function:

\[
8x^2 + 8x + 2 + 4 = 8x^2 + 8x + 6
\]

Thus, the fully expanded expression for \( f(2x + 1) \) is:

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

Would you like any further details on this problem or have any questions?

Here are 5 related questions:
1. How do we find \( f(x) \) if the function involves cubic terms?
2. What happens if the function involves complex numbers like \( f(z) = z^2 + iz \)?
3. How can we apply this method to trigonometric functions, e.g., \( f(\sin(x)) \)?
4. What if we differentiate or integrate \( f(2x + 1) \) with respect to \( x \)?
5. How do we find the inverse of a quadratic function like \( f(z) \)?

**Tip**: When expanding binomials like \( (a + b)^2 \), use the identity \( (a + b)^2 = a^2 + 2ab + b^2 \) to simplify the process.

Learn how to evaluate and fully expand the function f(z) = 2z^2 + 4 for f(2x + 1). This problem covers key algebraic concepts such as function evaluation, quadratic expansion, and binomial expansion. Suitable for students in Grades 9-11.

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