# What is the domain and range of a sine graph?

Jul 26, 2018

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

$f \left(x\right) = A \sin \left(B x + C\right) + D$

Where

• $A = \text{Amplitude}$
• $2 \pi / B = \text{Period}$
• $- C / B = \text{Phase shift}$
• $D = \text{Vertical shift}$

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 $\mathbb{R}$.

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 $\sin x$ is $1$, while its minimum is $- 1$.

Hence:

$\text{Range" = [D-A, A+D] or "Range} = \left[A + D , D - A\right]$

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

$\left[a , b\right] = \left[b , a\right]$

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

As a conclusion,

$f : \mathbb{R} \to \left[D - A , A + D\right]$

Jul 26, 2018

$\text{ }$
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:

$\text{ }$
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.