Range in Multivariable Calculus | Introduction to Range.

Range in Multivariable Calculus |

Introduction to Range:

Range is all the values that come out from the Function.

The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.

How to find the range

• The range of a function is the spread of possible y-values (minimum y-value to maximum y-value)
• Substitute different x-values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? Or maybe not equal to certain values?)
• Make sure you look for minimum and maximum values of y.

f(x) = x² + 2

is defined for all real values of x (because there are no restrictions on the value of x).

Hence, the domain of $\displaystyle f{{\left({x}\right)}}$ is

"all real values of x".

Range: Since x² is never negative, x² + 2 is never less than $\displaystyle{2}$

Hence, the range of $\displaystyle f{{\left({x}\right)}}$ is

"all real numbers $\displaystyle f{{\left({x}\right)}}\ge{2}$".

" y = sin x "
the value of sin x must always be between −1 and 1. So "the domain of f(x) = sin x contains all the real numbers",
but" the range is −1 ≤ sin x ≤ 1". We can also see that the function repeats itself every 360◦ . We can say that sin x = sin(x + 360◦ ). We say the function is periodic, with periodicity 360◦ .