Why is f not a function from R to R if

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When solving exercises related to functions, a common type of problem is to find out if what we are given is actually a function or not.

As with almost everything in mathematics, we need to start with the definition.

Definition: “Let A and B be nonempty sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f:A → B.” Discrete mathematics and its applications By Rosen.

Now, we have all we need to solve this type of exercise.

Table of Contents

a) f(x)=1/x?

Because we have to determine why f is not a function from R to R, we just need to show that f doesn’t hold the definition of a function.

So, the sets A and B equal R, or the set of real numbers.

What will be the value of f(0)?

Division by 0 is not defined. Se we can state that f(0) is not defined.

Therefore, f is not a function as each element of A should have a corresponding value of B.

b) f(x)=√x?

Again, we should follow the definition of a function to show why f is not a function.

In this case, we know that x can take any value of R, specifically, negative values.

We also know that the square root of negative values is not defined on the set of real numbers.

Therefore, f is not a function because is not defined for x<0.

c) f(x)=±√(x2+1)?

In this case, we can see that for each value of x, f can take two values, one positive and one negative.

When we examine the definition of a function, it reads “A function f from A to B is an assignment of exactly one element of B to each element of A”.

Therefore, f is not a function because it assigns two elements of B to each element of a.

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