Question

which of the following statements is not true?
choose the correct answer.
a. every function that passes the vertical line test is one - to - one.
b. a function f is one - to - one if for any two range values f(u) and f(v), f(u)=f(v) implies that u = v.
c. a function f is one - to - one if for any values a≠b in the domain of f, f(a)≠f(b).
d. every function that passes the horizontal line test is one - to - one.

Explanation:

• Option A:
• The vertical line test is used to determine if a relation is a function. A function passes the vertical line test if for every \(x\)-value there is exactly one \(y\)-value. But a function like \(y = x^{2}\) passes the vertical line test (it is a function) but is not one - to - one (since \(f(2)=f(- 2)=4\)).
• Option B:
• By the definition of a one - to - one function, if \(f(u)=f(v)\) implies \(u = v\) for \(u\) and \(v\) in the domain of \(f\), then \(f\) is one - to - one.
• Option C:
• If \(a

eq b\) in the domain of \(f\) and \(f(a)
eq f(b)\), this is another way of stating the one - to - one property. If different inputs give different outputs, the function is one - to - one.

• Option D:
• The horizontal line test: A function \(y = f(x)\) is one - to - one if and only if every horizontal line intersects the graph of the function at most once. So, if a function passes the horizontal line test, it is one - to - one.

Answer:

A. Every function that passes the vertical line test is one - to - one.