What is the domain and range of y = cos | x | ?

Question

What is the domain and range of       y    =    cos          |      x      |      ?

Tianna Cameron · Accepted Answer

This is a fun problem because we are presented with a modified version of the

\cos (x)

function, but as we will see, it is, in fact, not in any way different from the standard version.

\cos | x |

is the

\cos (x)

function with the absolute value of

x

input into it. What this means is that if

x < 0

then it is replaced with

x

. If

x < 0

then it is replaced with

- x

.
This means that our function is actually a two part piecewise function:

f (x) = {\begin{cases} \cos | x | = \cos (x) if x \geq 0 \\ \cos | x | = \cos (- x) if x < 0 \end{cases}

However, cos is also an even function. For an even function,

f (- x) = f (x)

.
This means that:

f (x) = {\begin{cases} \cos | x | = \cos (x) = \cos (x) if x \geq 0 \\ \cos | x | = \cos (- x) = \cos (x) if x < 0 \end{cases}

.
The domain and range will be precisely the same as for the original function, that is:

x

can be all real numbers, and

y

will range from

- 1

to

1

.

What is the domain and range of y = cos|x|?