Determine domain and range of linear functions

The linear equation $y=mx+c$y=mx+c performs a certain task. It takes any value of $x$x and develops from it a unique value of $y$y. For example, given the line $y=4x-7$y=4x−7 a value of $x$x as, say $3$3, generates a value of $y$y as $5$5.

In mathematics, we refer to $y$y as being a function of $x$x, so that $y=f\left(x\right)$y=f(x).  We say that the function $y=4x-7$y=4x−7 maps $x=3$x=3 to $y=5$y=5. A short hand notation is to write that since $f\left(x\right)=4x-7$f(x)=4x−7, then $f\left(3\right)=5$f(3)=5.

The numbers that are being mapped - the $x$x values - are said to belong to a set called the domain. When sketching the line $y=mx+c$y=mx+c, with non-zero slope $m$m and $y$y- intercept $c$c, we normally take the Domain to be all real numbers, so that the line is continuously drawn infinitely in both directions. When this happens we say that the domain is given by the open interval $\left(-\infty,\infty\right)$(−∞,∞). In fact, this is the natural domain of the linear function. 

The numbers that the $x$x values are being mapped to - the $y$y values - are said to belong to a set called the range. For linear functions with a non-zero slope, if the domain is the natural domain given by $\left(-\infty,\infty\right)$(−∞,∞), then the range will also be every real number $\left(-\infty,\infty\right)$(−∞,∞). That is to say, any real number $y$y can be obtained by mapping the appropriate $x$x value.

This is why the continuous line not only extends indefinitely in both left and right directions, but also extends indefinitely up and down. 

There are however linear functions with a range comprising of just a single number even though the function is defined over the natural domain. For example the linear function $y=4$y=4 is drawn as a line parallel to the $x$x -axis, so that every value of $x$x is mapped to the single value $4$4. In this case the domain is the open interval $\left(-\infty,\infty\right)$(−∞,∞), but the range is simply $\left\{4\right\}${4}.

Note however that the line $x=4$x=4, which is a line parallel to the $y$y- axis, is strictly speaking not a function. As stated in the first paragraph, a function develops for each value of $x$x a unique value of $y$y. In this specific case, the single value $x=4$x=4 is being mapped onto an infinite number of $y$y values. Even though the equation does not represent a function, we can still say that the domain is the set consisting of the single number $\left\{4\right\}${4}, and the range is given as $\left(-\infty,\infty\right)$(−∞,∞).

In general, a mapping of $x$x values to $y$y values is called a relation, and a relation which maps $x$x values to a unique y value is called a function. 

Note that every linear function with a non-zero slope must have exactly one $x$x- intercept. The value of $x$x which makes $y=0$y=0 is called a zero of a function. For example the function $y=-4x+20$y=−4x+20 has the single zero given by $x=5$x=5, simply because that value of x is mapped to the number 0 in the range.   

Worked Examples

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