The plots of two functions f and g are shown below.

Which of the following statements describes the relationship between f and g?

Question

The plots of two functions f and g are shown below.

Which of the following statements describes the relationship between f and g?

BlackTom AI · Accepted Answer

The question asks to describe the relationship between the two plotted functions f and g, where f is shown in orange and g in blue. Because the provided answer options are missing in the prompt, I will first restate what we would typically evaluate in such a scenario, and then discuss what each potential claim would imply.

- If the claim were that g is an antiderivative of f (i.e., g' = f): this would mean that the derivative of g at any x equals the value of f at that x. Visually, this requires us to see that the slope (steepness) of the g-curve at each point x matches the corresponding value of f(x). Since we only have the two entire graphs shown simultaneously, we cannot directly read off derivatives from g’s graph alone without additional slope information or tangent lines. Therefore, without explicit slope data or a demonstration that the tangent of g corresponds exactly to f, this claim cannot be confirmed from the plot alone.

- If the claim were that f is an antiderivative of g (i.e., f' = g): this would require the derivative of f to equal g at each x. Again, visually assessing this is challenging because we would need to compare the slope of f at every x to the value of g at that x. The two curves’ shapes alone do not provide a straightforward one-to-one correspondence between f's derivative and g across all x, so this cannot be verified from the graphs alone.

- If the claim were that f and g are simply related by being each other’s derivatives or antiderivatives in some restricted interval, we would need explicit evidence (e.g., matching tangent slopes with opposite curve values, or a consistent derivative-curve correspondence across the domain). The plot as shown does not supply such explicit evidence; the shapes alone do not establish a universal derivative relationship.

- If the claim were that f and g are independent or only coincident at certain points, we would need to check at those specific x-values where the two curves intersect or share particular properties. An isolated intersection does not imply a derivative relationship across the entire domain.

Because the actual list of answer choices is missing, I cannot assess which specific statement is correct or incorrect. The key idea is that to declare g is an antiderivative of f, one must verify that g' equals f across the domain, which requires slope information from g or a known functional relationship; simply inspecting the plotted curves without derivative data does not suffice to confirm or deny this relationship.

In summary, with only the two graphs and no explicit answer options provided, we cannot definitively determine whether g is an antiderivative of f or any other claimed relation. The proper evaluation would rely on comparing derivatives (slopes) of one graph to the values of the other, which is not directly readable from a static plot without additional tangent information.

MATH_1225_17243_202501 4.9 Antiderivatives

The plots of two functions f and g are shown below. Which of the following statements describes the relationship between f and g?

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