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Functions: basic concepts

A function $$f:X\rightarrow Y$$ maps each input element $$x$$ of a given set $$X$$ to exactly one value $$y$$ of a set $$Y$$.

An element of $$X$$ cannot be mapped to more than one element of $$Y$$, but two elements of $$X$$ may be mapped to the same element of $$Y$$.

$$f(x) = \vert x\vert$$ associates both $$1$$ and $$-1$$to $$1$$. This is a function. The mapping that sends $$1$$ to both $$-1$$ and $$1$$ is not a function.

The value $$f(x)$$ is the image of $$x$$. The image of $$x$$ is often written as $$y$$. This image is expressed by $$$f:x\mapsto f(x).$$$

The set for which the function is defined is the domain. It is usually denoted by $$D_f$$. Often the function is written as $$f:X\rightarrow Y$$ where the set $$X$$ contains the domain. This is usually done for convenience.

The domain of the function $$f(x) = 1/x$$ is $$D_f =\R\backslash\{0\}$$.

The collection of values $$f(x)$$ for all the elements $$x\in D_f$$ is the range of $$f$$. This is contained in the set $$Y$$, which is called the target set or co-domain.

$$f(x)= 1/(x-1)^2$$ is defined for $$x\ne 1$$. Its range is $$(0,+\infty)$$.

 The function $$x\mapsto 1/(x-1)^2.$$
The function $$x\mapsto 1/(x-1)^2.$$
The function is defined only on its domain. The resulting values give the range. The co-domain could be all of the $$y$$-axis.
The function is defined only on its domain. The resulting values give the range. The co-domain could be all of the $$y$$-axis.

A function $$f$$ is periodic with period $$T\gt 0$$ if $$$f(x+T)=f(x)$$$ for all $$x$$. The period of a periodic function is the smallest positive $$T$$ with this property.

The function $$\sin x$$ is periodic with period $$2\pi$$. The function $$\tan x$$ is periodic with period $$\pi$$. The function $$x$$ is not periodic. A constant function is periodic for any period.

The graph of a periodic function is invariant by translation along the $$x$$-axis of distance $$T$$.

In practice, it is enough to define a periodic function on an interval of width $$T$$. For any $$x\in\R$$, we can find an integer $$k$$ so that $$x-kT$$ lies in the interval on which the function was defined. We then have $$f(x) = f(x-kT)$$.

All these waves have the same period.
All these waves have the same period.

The graph of a function is the representation of the function using the Cartesian coordinate system. It is the collection of all points $$$\big(x,f(x)\big)$$$ where $$x$$ ranges through $$D_f$$.

 The graph of $$x\mapsto 1/(x-1)^2.$$
The graph of $$x\mapsto 1/(x-1)^2.$$

Because $$f(x)$$ takes only one value, each $$x$$-coordinate in the graph has only one corresponding $$y$$-coordinate. Any vertical line intersects the graph at most once.

The blue circle on the left is not the graph of a function since the red vertical line intersects it twice. The semi-circle on the right is the graph of a function.
The blue circle on the left is not the graph of a function since the red vertical line intersects it twice. The semi-circle on the right is the graph of a function.

A numerical function is even if its graph is symmetric across the $$y$$-axis. Analytically, this means that, for all $$x$$, $$$ f(-x) = f(x).$$$

Examples of even functions are $$x^2$$, $$\cos(x)$$, $$\vert x\vert$$.

A numerical function is odd if its graph has rotational symmetry of angle $$\pi$$ about the origin. Analytically, this means that, for all $$x$$, $$$ f(-x) = - f(x).$$$

Examples of odd functions are $$x$$, $$x^3$$, $$\sin(x)$$ and $$\tan(x)$$.

Parity is the property of being either odd or even for a function.

Some functions are neither even nor odd, such as $$x^2 - 3$$

A function $$f$$ is bounded if, for some constant $$M$$ and all $$x\in D_f$$, $$$\vert f(x) \vert \le M.$$$ It is bounded below if $$f(x)\ge -M$$ and bounded above if $$f(x)\le M$$.

The graph of a bounded function lies within two horizontal lines. The graph of a function bounded below remains above an horizontal line.

$$\sin(x)$$ is bounded; $$x^2$$ is bounded below; $$x^3$$ is unbounded.

A function is increasing if, whenever $$x\gt y$$, $$$f(x)\ge f(y).$$$ It is decreasing if $$f(x)\le f(y)$$. A function is strictly increasing or strictly decreasing if the inequalities are all strict. A function is monotonic if it is either always increasing or always decreasing.

The graph of an increasing function "climbs up or stays flat" as $$x$$ increases.

$$x^3$$ is strictly increasing; $$e^{-x}$$ is strictly decreasing; $$\max(x,0)$$ is increasing, but not strictly increasing when $$x\lt 0$$.