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:

$\frac{\pi}{2b}+\frac{c}{b}$π2`b`+`cb`	$=$=	$\frac{\pi}{2b}+\frac{2c}{2b}$π2`b`+2`c`2`b`
	$=$=	$\frac{\pi+2c}{2b}$π+2`c`2`b`

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.

Careful!

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.

Remember!

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.
D
Select 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 $\left(-\infty,\infty\right)$(−∞,∞).
A
Multiplying $x$x by $2$2 translates a function vertically.
B
Multiplying $x$x by $2$2 changes the position of the asymptotes.
C
Multiplying $x$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.

Loading Graph...

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.
A
All real $x$x.
B
All real $x$x except when $x=\pi k+\frac{\pi}{2}$x=πk+π2 for all integer $k$k.
C
All real $x$x except when $x=\pi k+\frac{\pi}{6}$x=πk+π6 for all integer $k$k.
D
State 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.
A
All real $x$x except when $x=2\pi k+\pi$x=2πk+π
B
All real $x$x except when $x=\pi k+\frac{\pi}{2}$x=πk+π2 for all integers $k$k.
C
All real $x$x except when $x=\pi k+\frac{\pi}{2}-3$x=πk+π2−3 for all integers $k$k.
D
State 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: $\left(\editable{},\editable{}\right)$(,)

Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

Domain and range of tangent curves