Contradiction in Inequality: No Real Solution
Assume that there exists a real number \(x\) such that \(x^2 + 1 < 0\). Use proof by contradiction to show that this assumption leads to a logical impossibility.
Graphical Inequality Analysis
Consider the graphs of two functions: a parabola \(f(x)\) and a straight line \(g(x)\). Their graphs intersect at the points where \(x = 1\) and \(x = 3\). Assume that for \(1 < x g(x)\). Provide your answer in interval notation.