Short Answer 
This answer explains the definition of a function and introduces the vertical line test, which determines if a graph represents a function by checking if a vertical line intersects the graph more than once. Graphics a, b, d, and e pass the test and qualify as functions, while options that fail the test do not.
Step 1: Understand the Definition of a Function
The definition of a function states that for each value of the input variable x, there must be only one corresponding output value y. This means that if you pick any value along the x-axis, you should be able to find a unique value on the y-axis. A useful tool to determine if a graphical representation is a function is the vertical line test.
Step 2: Apply the Vertical Line Test
The vertical line test assesses whether a graph represents a function by checking if a vertical line drawn through any point on the x-axis intersects the graph at more than one point. If it does, that section of the graph does not represent a function. For example, when applying this test to graphics, you find that options c and f fail the test because there are multiple y values for certain x values.
Step 3: Identify Functions Among the Graphics
Based on the vertical line test and the defined characteristics of functions, we can determine which graphics are functions. The graphics that pass this test are listed as follows:
- Graphic a: Constant line, hence a function.
- Graphic b: Single y value for each x, hence a function.
- Graphic d: Only one y for each x, hence a function.
- Graphic e: Each x corresponds to a single y, hence a function.
Thus, graphics a, b, d, and e represent functions.