Question

What is the domain and range of a sine graph?

Answer 1

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]`

Answer 2

`" "`
Domain:

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

Interval Notation: `color(green)((-oo, oo)`

Range:

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

Interval Notation: `color(green)([-1, 1]`

Explanation:

`" "`
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.