The natural domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of values that can be plugged into the function to get a real number as the output.
To find the natural domain of a function, you need to look for any restrictions on the input values. This includes avoiding division by zero, taking the square root of a negative number, or any other operation that would result in an undefined or imaginary number. Once you have identified these restrictions, you can determine the set of valid input values.
The range of a function is the set of all possible output values (y-values) that the function can produce. In other words, it is the set of values that the function can take on as its output for different input values.
To find the range of a function, you can either graph the function and look at the y-values that are covered, or you can use algebraic methods. Algebraically, you can set up and solve equations to determine the possible output values for different input values. However, it is important to note that not all functions have a range that can be easily determined.
Yes, it is possible for the natural domain and range of a function to be the same. This can happen when the function is defined for all real numbers and its output values cover all real numbers as well. In this case, the natural domain and range would both be the set of all real numbers, denoted as (-∞, ∞).