The properties of a function

A function gives every $\displaystyle x$ one $\displaystyle y$, and its graph shows its properties at a glance. You read them from the graph:

• the intersection points with the axes;
• the positive and negative domains;
• the extremum points – maximum and minimum;
• the increasing and decreasing domains.

The graph of the function $\displaystyle f$ is made of straight pieces between the points $\displaystyle (-5,-4)$, $\displaystyle (-3,4)$, $\displaystyle (0,-2)$, $\displaystyle (2,2)$ and $\displaystyle (5,-4)$.
At both ends it goes on beyond the grid, downwards.

• With the $\displaystyle x$-axis: the points where $\displaystyle {y}={0}$. For $\displaystyle f$: $\displaystyle (-4,0)$, $\displaystyle (-1,0)$, $\displaystyle (1,0)$, $\displaystyle (3,0)$.
• With the $\displaystyle y$-axis: the point where $\displaystyle {x}={0}$. For $\displaystyle f$: $\displaystyle (0,-2)$.

The function is positive where its graph is above the $\displaystyle x$-axis, and negative where it is below it.

• Positive: $\displaystyle -4<x<-1$, $\displaystyle 1<x<3$.
• Negative: $\displaystyle x<-4$, $\displaystyle -1<x<1$, $\displaystyle x>3$.

An extremum point is a peak or a valley of the graph. A maximum point (a peak) is higher than all the points of the graph near it; a minimum point (a valley) is lower than all the points near it.

• Maximum points of $\displaystyle f$: $\displaystyle (-3,4)$, $\displaystyle (2,2)$.
• Minimum point of $\displaystyle f$: $\displaystyle (0,-2)$.

An extremum point is an ordered pair. Its $\displaystyle y$ is the maximum (or minimum) value: at the point $\displaystyle (-3,4)$ the maximum value of $\displaystyle f$ is 4, and it is reached at $\displaystyle {x}={-3}$.

An absolute maximum is the highest point of the whole graph; a local maximum is the highest only near itself. The same goes for a minimum.

• $\displaystyle (-3,4)$ is an absolute maximum point of $\displaystyle f$: no point of the graph is higher.
• $\displaystyle (2,2)$ is a local maximum point: the point $\displaystyle (-3,4)$ is higher.
• $\displaystyle (0,-2)$ is a local minimum point. $\displaystyle f$ has no absolute minimum: both its ends go down without end.

On this graph, $\displaystyle A(-2,-4)$ is a local minimum point and $\displaystyle B(2,4)$ is a local maximum point. Neither is absolute: on the left the graph rises without end, and on the right it falls without end.

A function may have no extremum point at all: the line $\displaystyle {y}={2x+1}$ rises all the time.

A function is increasing where its graph rises as $\displaystyle x$ grows (going right), and decreasing where it falls. The domains are written with $\displaystyle x$:

• $\displaystyle f$ is increasing for $\displaystyle x<-3$ and for $\displaystyle 0<x<2$.
• $\displaystyle f$ is decreasing for $\displaystyle -3<x<0$ and for $\displaystyle x>2$.

The graph of the function $\displaystyle g$ is made of straight pieces between the points $\displaystyle (-4,4)$, $\displaystyle (-2,-4)$, $\displaystyle (1,2)$, $\displaystyle (3,-2)$ and $\displaystyle (5,2)$; at both ends it goes on upwards.
Mark on the graph the intersection points of $\displaystyle g$ with the axes, and its extremum points.
Write:

• its positive and negative domains;
• for each extremum point: maximum or minimum, local or absolute;
• its increasing and decreasing domains;
• the minimum value of $\displaystyle g$.

Sketch the graph of a function with all these properties:

• it meets the $\displaystyle x$-axis only at $\displaystyle (-3,0)$ and $\displaystyle (2,0)$, and the $\displaystyle y$-axis at $\displaystyle (0,4)$;
• it has a maximum point $\displaystyle (-1,5)$, and no minimum point.

A function is increasing for $\displaystyle x<0$ and for $\displaystyle x>3$, and decreasing for $\displaystyle 0<x<3$.
Its maximum value is 2, and its minimum value is $\displaystyle -4$.
Sketch it, and write its extremum points.

Answer, and explain.

• A function has a minimum point $\displaystyle (2,-5)$. What is its minimum value, and at which $\displaystyle x$ is it reached?
• Can a local maximum value of a function be smaller than a local minimum value? Show it in a sketch.
• Can the graph of a function meet the $\displaystyle y$-axis at two points?