Suppose $f(x)$ is a function that has this property:
For all real numbers \(a\) and \(b\) such that $a \(y=f(x)\) between \(x=a\) and \(x=b\) lies below the line segment whose endpoints are \((a,f(a))\) and \((b,f(b))\).
(A function with this property is called strictly~convex.)
Given that $f(x)$ passes through $(-2,5)$ and $(2,9)$, what is the range of all possible values for $f(1)$? Express your answer in interval notation.
Let a=-2 and b=2. Then the special property of f(x) tells us that the point (1,f(1)) lies below the line segment through (-2,5) and (2,9). This line segment is part of the line with equation y=x+7, so it passes through (1,8). Therefore, f(1)<8. There are no further restrictions on f(1). Thus, as an interval, the range of possible values for f(1) is \(\boxed{(-\infty,8)}\)
