Given the following functions, their graphs and the composite functions represented graphically, verify they are correct representations using derivative rules. Consider the functions: f(x) = 3x^2 + 2x + 1 g(x) = 4x^3 - x +5 (I:1)

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Let’s parse the question and what each option is claiming about the derivative representations of combinations of f and g.
Option a states: k(x) = f(x) − g(x) is correctly represented by the difference of f(x) and g(x).
- This is a standard linear combination: subtracting the two functions pointwise. The derivative rules confirm that the derivative of k is k'(x) = f'(x) − g'(x), which aligns with how a difference of functions behaves. There’s nothing contradictory here, so this representation is valid.
Option b states: h(x) = f(x) + g(x) is correctly represented as the sum of f(x) and g(x).
- This is the classic sum of two functions. The derivative of h is h'(x) = f'(x) + g'(x). Graphically and algebraically, adding the functions corresponds to adding their outputs for every x, so this representation is correct as well.
Option c states: All of the above are correct.
- If both a and b are correct, then this option would also be correct by logical necessity. However, to verify c independently you’d need to confirm both a and b hold, which we did above, so conceptually c follows from the correctness of a and b.
Option d states: m(x) = 2f(x) is correctly represented by doubling f(x).
- Doubling a function means scaling its outputs by 2: m(x) = 2 f(x). The derivative would be m'(x) = 2 f'(x). This is a standard linear scaling property of derivatives and is consistent with how function doubling behaves graphically and analytically.
In summary:
- Option a is correct because the difference of two functions is represented by subtracting their values pointwise, and its derivative follows the difference rule.
- Option b is correct because the sum of two functions is represented by adding their values pointwise, with the derivative following the sum rule.
- Option d is correct because multiplying a function by a constant (here 2) scales its graph accordingly, and the derivative scales by the same constant.
- Option c would be correct only if both a and b are correct, and given the standard definitions, they are, so c is a restatement that aligns with a and b.
Thus, each individual statement about basic algebraic composition and derivative rules aligns with fundamental calculus principles, and together they describe standard, valid representations of composed or scaled functions.

Given the following functions, their graphs and the composite functions represented graphically, verify they are correct representations using derivative rules. Consider the functions: f(x) = 3x^2 + 2x + 1 g(x) = 4x^3 - x +5 (I:1)

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