Question

The graph of a function never has two different points with the same x-coordinate because.. A. the graph of a function cannot be a straight line. B. each input value is mapped to a single output value. C. each input value is mapped to more than one output value. D. the graph of a function is a vertical line.

Answer 1

Answer: The correct reason is (B) each input value is mapped to a single output value.

Step-by-step explanation: The given statement is:

"The graph of a function never has two different points with the same x-coordinate."

We are to select the correct reason for the above statement.

The definition of a FUNCTION is as follows:

In simple language, a function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output.

That is, if y = f(x) is a function of a variable 'x', then we have

`x_1=x_2~~~~\Rightarrow f(x_1)=f(x_2).`

(A) The first statement is "the graph of a function cannot be a straight line."

The statement is false. For example, let us consider y = x + 2. Obviously, it is a function because every value of 'x' will result in only one value of 'y'.

Also, the graph here will be a straight line as shown in the attached figure (1).

(B) The first statement is "each input value is mapped to a single output value."

The statement is true according to the definition of a function.

(C) The third statement is "each input value is mapped to more than one output value."

The statement is false by the definition of a function.

(D) The fourth statement is "the graph of a function is a vertical line."

The statement is false. For example, if we consider the vertical line x = 2. Then, we see from the attached figure (2) that the points (x, y) = (2, 1), (2, 0), (2, 3), etc. all lie on the line.

This shows that one input value leads to infinitely many values of 'y'. Hence, this vertical line cannot be the graph of a function.

Thus, (B) is the correct statement.

Answer 2

The best answer to the question that is being stated above would be letter B. The graph of a function never has two different points with the same x-coordinate because the input value of each of the points is mapped to a single output value in the graph.