Question

Explain the difference between f(2) and f(x) = 2. Use a verbal explanation and create examples to make it very clear. Please help me out

Answer 1

Let x be a variable, and f(x) a function of that variable.

f(2) is a number ; it's the value of the function f(x) when x = 2, assuming that value exists.

f(x) = 2 is a statement ; it says there is some value of x for which the function returns a value of 2. Because it's a statement, it can be true or false.

Here's an example:

Define f(x) = sin(x). Then f(2) = sin(2) ≈ 0.909297.

If x is a real number, then the statement sin(x) = 2 is false, because -1 ≤ sin(x) ≤ 1 for all real x.

If we replace 2 with 1, on the other hand, we get

f(1) = sin(1) ≈ 0.84147

and

sin(x) = 1  ==>  x = π/2 + 2nπ

where n is any integer. (So we're talking about numbers like π/2, -3π/2, 5π/2, -7π/2, and so on.) We're saying here that any real number x of the above form satisfies the equation and makes the statement true.