Is the vertex of an equation included in the increasing and decreasing intervals

Question

Is the vertex of an equation included in the increasing and decreasing intervals of the function? For examples, I have function $f(x) = |x-2|+1$. Would the decreasing and increasing intervals be written as $(-\infty,2]$ and $[2,\infty)$? Or would they be written as $(-\infty,2)$ and $(2,-\infty)$. Thanks

Answer 1

The graph of the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = |x - 2| + 1$ is shown below.

We note that its range is $[1, \infty)$ and that the minimum value of $1$ occurs when $x = 2$ since the expression $|x - 2| + 1 > |2 - 2| + 1 = 0 + 1 = 1$ for each $x \in \mathbb{R}$.

Let's consider the definitions of increasing and decreasing.

Definition. A function $f$ is said to be increasing on an interval $I$ if for each $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) \leq f(x_2)$. A function $f$ is said to be strictly increasing on an interval $I$ if for each $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) < f(x_2)$.

If $x \geq 2$, then $|x - 2| = x - 2 \geq 0$, so $$f(x) = |x - 2| + 1 = x - 2 + 1 = x - 1$$ which is strictly increasing on $[2, \infty)$ since whenever $2 \leq x_1 < x_2$, $2 - 1 \leq x_1 - 1 < x_2 - 1$. In particular, if $x > 2$, then $f(2) = 2 - 1 = 1 < x - 1 = f(x)$.

Definition. A function $f$ is said to be decreasing on an interval $I$ if for each $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) \geq f(x_2)$. A function $f$ is said to be strictly decreasing on an interval $I$ if for each $x_1, x_2 \in I$ with $x_1 < x_2$, $f(x_1) > f(x_2)$.

If $x \leq 2$, then $|x - 2| = -(x - 2) = -x + 2 \geq 0$, so $$f(x) = |x - 2| + 1 = -x + 2 + 1 = -x + 3$$ which is strictly decreasing on $(-\infty, 2]$ since whenever $x_1 < x_2 \leq 2$, $-x_1 + 3 > -x_2 + 3 > -2 + 3$. In particular, if $x < -2$, then $f(x) = -x + 3 > 1 = -2 + 3 = f(2)$.

Note that it is the $x$-coordinate of the vertex of the graph of the function that is included in the intervals where the function is increasing or decreasing, not the vertex. As Jean Marie pointed out in the comments, the vertex is of the graph of the equation $y = |x - 2| + 1$, not the equation itself.

You are using interval notation incorrectly. Let's look at definitions. \begin{align*} (a, b) & = \{x \in \mathbb{R} \mid a < x < b\}\\ [a, b] & = \{x \in \mathbb{R} \mid a \leq x \leq b\}\\ [a, b) & = \{x \in \mathbb{R} \mid a \leq x < b\}\\ (a, b] & = \{x \in \mathbb{R} \mid a < x \leq b\}\\ (a, \infty) & = \{x \in \mathbb{R} \mid x > a\}\\ [a, \infty) & = \{x \in \mathbb{R} \mid x \geq a\}\\ (-\infty, b) & = \{x \in \mathbb{R} \mid x < b\}\\ (-\infty, b] & = \{x \in \mathbb{R} \mid x \leq b\} \end{align*} Thus, $(-\infty, 2) = \{x \in \mathbb{R} \mid x < 2\}$, while $(2, -\infty) = \{x \in \mathbb{R} \mid 2 < x < -\infty\} = \emptyset$. The smaller number must be written first, otherwise the interval is empty.