Question

Which of the following graphs represent the function of x?
(A) (B)
(C)(D)
Choose the correct answer from the options given below:

• A. (A), (B) and (C) only
• B. (A), (B) and (D) only
• C. (A) and (D) only
• D. (A) and (C) only

Answer 1

The Correct Option is C

To determine which graphs represent a function of x, we need to understand the definition of a function. A function is a relation in which each input (or x-value) corresponds to exactly one output (or y-value).

The vertical line test is a useful tool for this: any vertical line drawn through the graph of a function should intersect the graph at most once.

Let's evaluate the given graphs:

1. Graph (A): This graph depicts two linear arms forming a "V" shape, which is the absolute value function y = |x|. Any vertical line will intersect it at exactly one point. Thus, it is a function.
2. Graph (B): This resembles a sinusoidal wave that oscillates along the x-axis. Any vertical line through this graph can intersect it at multiple points, depending on the cycle, which means it is not a function.
3. Graph (C): This graph also shows a sinusoidal wave, but displaced along the y-axis. Similar to (B), a vertical line can intersect it more than once at many places. Thus, it is not a function.
4. Graph (D): This graph is a circle described by (x^2 + y^2 = r^2). A vertical line will intersect this circular graph at two points, failing the vertical line test. Technically, a circle itself isn’t a function of x, but considering if only part of it (like a semi-circle) is considered, one can argue about its function-like nature in certain scenarios.

Therefore, the correct representations of a function in terms of x are only graph (A) and graph (D) in specific applications.

Hence, the correct answer is: (A) and (D) only.