Question

Give an example of a function that is one-to-one on the entire real number line.

Short answer

Question: Provide an example of a function that is one-to-one on the entire real number line. Answer: The identity function, f(x) = x, is an example of a one-to-one function on the entire real number line.

Step-by-step solution

Understanding One-to-One Functions

A function is one-to-one (injective) if for every x1 and x2 in the domain, if f(x1) = f(x2), then x1 = x2. In other words, no two different input values will produce the same output value.

Choose a One-to-One Function

An example of a one-to-one function on the entire real number line is the identity function, f(x) = x. This function simply returns the input value as the output value.

Proof that the Function is One-to-One

To show that the function f(x) = x is one-to-one, we need to prove that for every x1 and x2 in the domain of real numbers, if f(x1) = f(x2), then x1 = x2. Let x1 and x2 be elements in the domain of real numbers. Suppose f(x1) = f(x2). Then, by the definition of the identity function, we have: f(x1) = x1 and f(x2) = x2 Since f(x1) = f(x2), we can write: x1 = x2 Thus, any two input values that produce the same output value are equal, which proves that the identity function f(x) = x is one-to-one on the entire real number line.