Question

State the domain and range of the following functions f(x) =1/x+3 g(x) =sqrt x+6

Answer 1

For the function
`f(x)=(1)/(x) +3`. The domain is
`\left(-\infty \:,\:0\right)\cup \left(0,\:\infty \:\right)` and the range is
`\left(-\infty, 3\right) \cup \left(3, \infty\right)`.

For the function
`g(x) =√(x+6)`. The domain is
`\left[-6, \infty\right)` and the range is
`\left[0, \infty\right)`.

Explanation:

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

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

`f(x)=(1)/(x) +3` is a rational function. A rational function is a function that is expressed as the quotient of two polynomials.

Rational functions are defined for all real numbers except those which result in a denominator that is equal to zero (i.e., division by zero).

The domain of the function is
`\left(-\infty \:,\:0\right)\cup \left(0,\:\infty \:\right)`.

The range of the function is
`\left(-\infty, 3\right) \cup \left(3, \infty\right)`.

`g(x) =√(x+6)` is a square root function.

Square root functions are defined for all real numbers except those which result in a negative expression below the square root.

The expression below the square root in
`g(x) =√(x+6)` is
`x+6`. We want that to be greater than or equal to zero.

`x+6\geq 0\\x\ge \:-6`

The domain of the function is
`\left[-6, \infty\right)`.

The range of the function is
`\left[0, \infty\right)`.