Question

Is `y = 5` an upper bound for `f(x) = x^2 + 5`?

Answer 1

No

Explanation:

By definition, a function `f(x)` is bounded above if there exists a real number `M in RR` such that the function value of `f(x) < M` for all `x in RR`
If a particular upper bound M is the lowest possible upper bound for a function, then it is called the supremum (sup).

Since `f(x)=x^2+5 >=5 AA x in RR`, it implies that `y=5` is not an upper bound for `f`.
In fact, `f` is not bounded above at all since it diverges to infinity.

However, 5 could be considered a lower bound for `f` and in fact, the greatest lower bound (infimum) of `f` since any value bigger than 5 is no longer a lower bound.

graph{x^2+5 [-47.8, 56.25, -10, 42]}