Finding the Domain and Range of Functions

If the inequality of a square root function has no solution,

(that is, no values of x for which the expression is less than zero), then the domain has no restrictions

If the rational function equation has no solutions,

(that is, no values of x make the denominator equal zero) then the domain has no restrictions

For square-root functions, the range is

any value of x such that x is greater than or equal to 0; it's not possible for a square root to result in a negative value, so functions in the form y = sqrt(x) or y = sqrt(ax + b) will always be greater than or equal to zero

When a function has no restrictions on its domain,

it is continuous from negative infinity to positive infinity along the x-axis; if the range also has no restrictions, then on one end, the graph tends towards negative infinity, and on the other end, it tends towards positive infinity

domain

the set of input values of a function or relation

range

the set of output values of a function or relation

The two common domain restrictions are

when there is an expression underneath a square root (or other even root), and when there is a denominator with a variable in it (as is the case with rational functions); this is because the expression under the even-root must be non-negative, and the denominator must not equal zero

With rational functions, the domain restrictions are

x-values that make the denominator equal to zero; set the denominator equal to zero and solve for x to find x-values to exclude from the domain

To find the domain and range of rational functions,

you need to consider domain restrictions because division by 0 leaves the function undefined

To find the domain and range of square root functions,

you need to consider domain restrictions because the square root of a negative value is undefined

It is easier to see the solutions of a rational function if

you rewrite the denominator in factored form