Determine o período e o conjunto imagem, construindo o gráfico de um período completo para cada função dada. a) por f(x) = 1+ 4 sen x. b) por f(x) = 2sen (x-3)

The given functions are

The sine function has a standard period of by definition. However, this might change if we use a factor as coefficient of the x-varible, but in this case we don't have that.

Therefore, the period of both trigonometric functions is .

Now, the images of each function is the y-variable set values that defines each function.

So, the function has an image defined by the set . It's impotant to notice that the range of a standard function is [-1,1], however, in this case, the function was shifted 1 unit up and it was streched by a factor of 4, that's why the standard image changes to .

About the second function , the image set is , because the function was streched by a factor of 2.

Additionally, the image attached shows the graph of the given functions.

answer: (yeees, you need a poet.)

//blooming serenity//

beautiful in the morning

mysterious at night

wonderous nature souring

with all it's mystical might

the flower that blooms

to the fox that looms

nocturnal creatures

with various features

a blooming flower soon in sight

as a new little hatchling fears to take flight  

afraid to leave it's calming nest,

but dear little bird, mother knows best

as sure as there are stars in the sky,

life is not about who, where what, when or why-

it's the journey that follows

adventures we swallow

of   nature's deep wonders,

natural disasters and storm's thunders.

blooming serenity

life and nature, truly is heavenly.

-kerstien lehman

step-by-step explanation: