Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$.
How to solve this problem?
Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$.
How to solve this problem?
Hint: if $x$ is quite large and positive, which of the two choices will be smaller? If $x$ is large and negative, which will be smaller? Can you find the crossover point?
Try graphing $y=x+6$ and $y=4-x$ together on one graph, then highlight or otherwise mark the parts of those graphs that make up $y=\min{x+6,4-x}$. The resulting shape should be a familiar type of basic function, perhaps translated and/or reflected.
HINT $\ $ For any continuous functions $\rm\:f,g,\:$ the intermediate value theorem implies that $\rm\ f-g\ $ will have constant sign between its roots. So you need only partition $\mathbb R$ by these roots and then evaluate the function at any test point of each interval to determine the sign on the whole interval.
In fact this decomposition technique generalizes to higher dimensions. It is known as the cylindrical decomposition algorithm - an effective interpretation of Tarski's decision procedure for the first-order theory of the reals.