The domain of a function is the set of all values that the independent variable (usually $x$x) can take and the range of a function is the set of all values that the dependent variable (usually $y$y) can attain.
Graphically speaking, we can determine the domain by observing the values of $x$x for which the function is defined over. We can also determine the range by observing the heights of each point on the graph.
Exploration
Consider the graph of $y=\tan x$y=tanx below.
$y=\tan x$y=tanx
Domain
Notice that the graph of $y=\tan x$y=tanx is undefined at periodic intervals of length $\pi$π. We state the domain as being:
All real values of $x$x, except where $x=k\pi+\frac{\pi}{2}$x=kπ+π2 for any integer $k$k.
More generally speaking, we can consider the function $y=a\tan\left(bx-c\right)+d$y=atan(bx−c)+d where $a,b,c,d$a,b,c,d are constants. Each constant applies a geometric transformation to the graph of $y=\tan x$y=tanx. Remember from our work on transformations, that the period is $\frac{\pi}{b}$πb, and the phase shift is $\frac{c}{b}$cb. So the first asymptote starts at:
| $\frac{\pi}{2b}+\frac{c}{b}$π2b+cb | $=$= | $\frac{\pi}{2b}+\frac{2c}{2b}$π2b+2c2b |
| $=$= | $\frac{\pi+2c}{2b}$π+2c2b |
and then they repeat every $\frac{\pi}{b}$πb units.
So for the general tan function, the domain is:
All real $x$x, except where $x=\frac{\pi k}{b}+\frac{\pi+2c}{2b}$x=πkb+π+2c2b for any integer $k$k.
That is quite an involved expression, but conceptually this is the same as:
The domain is all real values of $x$x, except for the asymptotes.
Notice that the constants $a$a and $d$d do not affect the domain of the function. This is because $a$a and $d$d relate to a vertical dilation and translation, which does not change the position of the vertical asymptotes.
Range
The graph of $y=\tan x$y=tanx has no minimums, maximums, horizontal asymptotes or holes. In other words, the range is all real values of $y$y or $\left(-\infty,\infty\right)$(−∞,∞). The only exception is when we restrict the graph to a smaller domain as shown below.
$y=\tan x$y=tanx over the domain $\left[0,\frac{\pi}{2}\right)$[0,π2)
In this case, the range is no longer all real values of $y$y but instead, $\left[0,\infty\right)$[0,∞). We include $y=0$y=0 in the range because the domain includes the value $x=0$x=0.
For a function of the form $y=a\tan\left(bx-c\right)+d$y=atan(bx−c)+d, where $a,b,c,d$a,b,c,d are constants:
- The domain is all real $x$x, except where $x=\frac{\pi k}{b}+\frac{\pi+2c}{2b}$x=πkb+π+2c2b for any integer $k$k
- The range is $\left(-\infty,\infty\right)$(−∞,∞)
However on a restricted domain, the range can be any subset of the real numbers.
Practice questions
question 1
Let $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=\tan2x$g(x)=tan2x.
Select the correct statement from the options below.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain and range.
A$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same range, but different domains.
B$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have different domains and ranges.
C$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain, but different ranges.
DSelect the most appropriate explanation for your answer to part (a).
Multiplying $x$x by $2$2 does not change the position of the asymptotes, and a vertical shift will not affect a range of $\left(-\infty,\infty\right)$(−∞,∞).
AMultiplying $x$x by $2$2 translates a function vertically.
BMultiplying $x$x by $2$2 changes the position of the asymptotes.
CMultiplying $x$x by $2$2 translates a function horizontally.
D
question 2
The graph of $f\left(x\right)=\tan\left(x-\frac{\pi}{3}\right)$f(x)=tan(x−π3) is shown.
Select the correct domain of $f\left(x\right)$f(x).
All real $x$x except when $x=\pi k+\frac{5\pi}{6}$x=πk+5π6 for all integer $k$k.
AAll real $x$x.
BAll real $x$x except when $x=\pi k+\frac{\pi}{2}$x=πk+π2 for all integer $k$k.
CAll real $x$x except when $x=\pi k+\frac{\pi}{6}$x=πk+π6 for all integer $k$k.
DState the range of $f\left(x\right)$f(x) using interval notation.
Range: $\left(\editable{},\editable{}\right)$(,)
question 3
Consider the function $f\left(x\right)=2\tan x+3$f(x)=2tanx+3.
Select the correct domain of $f\left(x\right)$f(x).
All real $x$x except when $x=\pi k+\frac{\pi}{2}+3$x=πk+π2+3 for all integers $k$k.
AAll real $x$x except when $x=2\pi k+\pi$x=2πk+π
BAll real $x$x except when $x=\pi k+\frac{\pi}{2}$x=πk+π2 for all integers $k$k.
CAll real $x$x except when $x=\pi k+\frac{\pi}{2}-3$x=πk+π2−3 for all integers $k$k.
DState the range of $f\left(x\right)$f(x) using interval notation.
Range: $\left(\editable{},\editable{}\right)$(,)
If we restrict the the graph of $f\left(x\right)$f(x) so that it's over $\left[0,\pi\right)$[0,π), what will the new range be?
New range: $\left(\editable{},\editable{}\right)$(,)