Level 7 - NCEA Level 2

Domain and range of tangent curves

Lesson

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 $x=k\times180^\circ+90^\circ$x=k×180°+90° 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{180^\circ}{b}$180°b, and the phase shift is $\frac{c}{b}$cb. So the first asymptote starts at:

$\frac{90^\circ}{b}+\frac{c}{b}$90°b+cb

$=$=

$\frac{90^\circ+c}{b}$90°+cb

and then they repeat every $\frac{180^\circ}{b}$180°b units.

So for the general tan function, the domain is:

All real $x$x, except where $x=\frac{k\times180^\circ}{b}+\frac{90^\circ+c}{b}$x=k×180°b+90°+cb 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 $\left[0^\circ,90^\circ\right)$[0°,90°)

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 $x=0^\circ$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 $x=\frac{k\times180^\circ}{b}+\frac{90^\circ+c}{b}$x=k×180°b+90°+cb 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 range, but different domains.
A
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have different domains and ranges.
B
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain, but different ranges.
C
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain and range.
D
Select the most appropriate explanation for your answer to part (a).
Multiplying $x$x by $2$2 translates a function horizontally.
A
Multiplying $x$x by $2$2 translates a function vertically.
B
Multiplying $x$x by $2$2 does not change the position of the asymptotes, and a vertical shift will not affect a range of $\left(-\infty,\infty\right)$(−∞,∞).
C
Multiplying $x$x by $2$2 changes the position of the asymptotes.
D

question 2

The graph of $f\left(x\right)=\tan\left(x-30^\circ\right)$f(x)=tan(x−30°) is shown.

Loading Graph...

Select the correct domain of $f\left(x\right)$f(x).
All real $x$x except when $x=k\times180^\circ+90^\circ$x=k×180°+90° for all integers $k$k.
A
All real $x$x except when $x=k\times180^\circ+120^\circ$x=k×180°+120° for all integers $k$k.
B
All real $x$x except when $x=k\times180^\circ+60^\circ$x=k×180°+60° for all integers $k$k.
C
All real $x$x.
D
State the range of $f\left(x\right)$f(x) using interval notation.

Range: $\left(\editable{},\editable{}\right)$(,)

question 3

Consider the function $f\left(x\right)=3\tan x+2$f(x)=3tanx+2.

Select the correct domain of $f\left(x\right)$f(x).
All real $x$x except when $x=k\times540^\circ+270^\circ$x=k×540°+270°
A
All real $x$x except when $x=k\times180^\circ+90^\circ$x=k×180°+90° for all integers $k$k.
B
All real $x$x except when $x=k\times180^\circ+88^\circ$x=k×180°+88° for all integers $k$k.
C
All real $x$x except when $x=k\times180^\circ+92^\circ$x=k×180°+92° 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 $\left[0^\circ,180^\circ\right)$[0°,180°), what will the new range be?

New range: $\left(\editable{},\editable{}\right)$(,)

Outcomes

M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

91257

Apply graphical methods in solving problems