Come up with an expression for a function f such that the domain of f is all real numbers, lim (f(x) = 3) as x approaches 2 and f(2) = 3.
To come up with an expression for a function f that satisfies the given conditions, we first need to determine the behavior of f around x = 2.
Since the limit of f(x) is defined as 3 as x approaches 2, it means that as x gets arbitrarily close to 2 from both sides, f(x) gets arbitrarily close to 3.
Furthermore, since f(2) is also equal to 3, this means that the function is continuous at x = 2.
To construct a function that satisfies these conditions, we can use the concept of a piecewise function. We will define one expression for x < 2 and another expression for x > 2.
Let's denote the function for x < 2 as f1(x) and the function for x > 2 as f2(x). Then, the expression for the function f(x) can be written as:
f(x) = { f1(x), x < 2
{ f2(x), x > 2
Since we want the limit of f(x) to be 3 as x approaches 2, we can set both f1(x) and f2(x) to 3. Therefore, the expression for f(x) becomes:
f(x) = { 3, x < 2
{ 3, x > 2
In summary, the expression for the function f(x) that satisfies the given conditions is f(x) = 3 for all real numbers.