What is the domain and range of a sine graph?

Let #f# be a generalized sinusoidal function whose graph is a sine wave:

#f(x)=Asin(Bx+C)+D#

Where

• #A = "Amplitude"#
• #2pi//B = "Period"#
• #-C//B = "Phase shift"#
• #D = "Vertical shift"#

The maximum domain of a function is given by all the values in which it is well defined:

#"Domain" = {x | x in RR and f(x) " is defined"}#

Since the sine function is defined everywhere on the real numbers, its set is #RR#.

As #f# is a periodic function, its range is a bounded interval given by the max and min values of the function. The maximum output of #sinx# is #1#, while its minimum is #-1#.

Hence:

#"Range" = [D-A, A+D] or "Range" = [A+D, D-A]#

The range depends on the sign of #A#. However, if we allow that

#[a,b] = [b,a]#

then the range is more simply defined as [D-A, A+D].

As a conclusion,

#f:RR -> [D-A, A+D]#

#" "#
Domain and Range of a SIN Graph:

Let us look at the SIN Graph first:

#color(blue)("Domain :"#

The domain of a function is the set of input values for which the function is real and defined.

#color(blue)((-oo < theta < oo)#

Domain restriction used for the SIN Graph to display ONE complete cycle.

#color(blue)("Range :"#

The set of output values (of the dependent variable) for which the function is defined.

As you can easily observe, the SIN graph goes up until #color(blue)(1# and goes down until #color(blue)(-1#

#color(blue)((-1 < theta < 1)#

Hope this helps.