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The question asks about the nature of the function \( f(x) \) given that \( f(x+2) - f(x) = 4x + 3 \) for all \( x \). Let's break this down step by step to identify the correct answer.

### 1. Understanding the Equation

The equation provided is:
\[
f(x+2) - f(x) = 4x + 3
\]
This equation gives the difference between the function values at \( x+2 \) and \( x \), which suggests something about the average rate of change of \( f(x) \) over an interval of length 2.

### 2. Analyzing the Equation for \( f(x) \)

We can rewrite the equation as:
\[
f(x+2) = f(x) + (4x + 3)
\]
This recursive form provides some information about how \( f(x) \) changes.

If \( f(x) \) were linear, then the difference \( f(x+2) - f(x) \) would be constant (not dependent on \( x \)). But here, \( 4x + 3 \) depends on \( x \), suggesting that \( f(x) \) is not linear.

### 3. Is \( f(x) \) Quadratic?

To check if \( f(x) \) could be quadratic, let's assume \( f(x) \) is a quadratic function, say \( f(x) = ax^2 + bx + c \). Then,
\[
f(x+2) = a(x+2)^2 + b(x+2) + c = a(x^2 + 4x + 4) + b(x + 2) + c = ax^2 + (4a + b)x + (4a + 2b + c)
\]
Now, calculate the difference \( f(x+2) - f(x) \):
\[
f(x+2) - f(x) = \left[ ax^2 + (4a+b)x + (4a + 2b + c) \right] - \left[ ax^2 + bx + c \right]
\]
\[
f(x+2) - f(x) = (4a+b)x + (4a + 2b + c - c) = (4a+b)x + 4a + 2b
\]
For this to match the given difference \( f(x+2) - f(x) = 4x + 3 \), we compare coefficients:
\[
4a + b = 4 \quad \text{and} \quad 4a + 2b = 3
\]
Solving this system of equations:
1. \( 4a + b = 4 \)
2. \( 4a + 2b = 3 \)

From the first equation, \( b = 4 - 4a \). Substitute this into the second equation:
\[
4a + 2(4 - 4a) = 3
\]
\[
4a + 8 - 8a = 3
\]
\[
-4a + 8 = 3
\]
\[
-4a = -5 \quad \Rightarrow \quad a = \frac{5}{4}
\]
Now substitute \( a = \frac{5}{4} \) back into \( b = 4 - 4a \):
\[
b = 4 - 4 \times \frac{5}{4} = 4 - 5 = -1
\]
So, \( a = \frac{5}{4} \) and \( b = -1 \).

Thus, \( f(x) \) is quadratic.

### 4. Conclusion

From the analysis, we conclude that \( f(x) \) is a quadratic function because the average rates of change over intervals of length 2 are constant. This matches option **C**.

### Answer: (C)

Would you like to explore more details on this, or do you have any further questions? Here are some additional questions for deeper understanding:

1. What happens if the difference \( f(x+2) - f(x) \) is a constant (independent of \( x \))?
2. How do we derive the general form of a quadratic function from its difference equation?
3. How would the solution change if \( f(x+2) - f(x) = cx \) for some constant \( c \)?
4. Can a cubic function ever have constant second differences like in this problem?
5. How does the second derivative test confirm the quadratic nature of a function?

Tip: When checking if a function is quadratic, compare its recursive form to the general quadratic formula, and use the differences to match terms.

The function f(x) satisfies f(x+2) - f(x) = 4x + 3 for all x. Which of the following is true?
(A) f(x) is a linear function because the average rates of change over consecutive intervals of length 2 are constant.
(B) f(x) is a linear function because the average rates of change over consecutive intervals of length 2 are linear.
(C) f(x) is a quadratic function because the average rates of change over consecutive intervals of length 2 are constant.
(D) f(x) is a quadratic function because the average rates of change over consecutive intervals of length 2 are linear.

Explore the nature of the function f(x) with a step-by-step breakdown of the equation f(x+2) - f(x) = 4x + 3. Learn how to determine whether the function is linear or quadratic based on its average rates of change. Suitable for students in Grades 10-12.

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