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Basic definitions

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.