We will discuss interval notation in greater detail later. In interval notation, we use a square bracket. We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. ![]() We also need to consider what is mathematically permitted. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. In this section, we will practice determining domains and ranges for specific functions. In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.įigure 1 Based on data compiled by 3 Finding the Domain of a Function Defined by an Equation In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Consider five major thriller/horror entries from the early 2000s- I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Horror and thriller movies are both popular and, very often, extremely profitable. Find the domain of a function defined by an equation.The range is given by U [1, + inf) which is the range of function f. Function f can take only two values: -1 and 2. The interval (-2, + inf) is open at x = -2 and the graph shows this with an "open point". In the interval (-2, + inf) the graph is a horizontal line y = f(x) = 2 (see formula for this interval above). Also this interval is closed at x = -2 and therefore the graph must show this : see the "closed point" on the graph at x = -2. In the interval (- inf, -2] the graph of f is a horizontal line y = f(x) = -1 (see formula for this interval above). We will graph it by considering the value of the function in each interval. The domain of f is the set of all real numbers. The above function is defined for all real numbers except for values of x in the interval (-2, 2] and x = 4.įind the domain and range of function f and graph it.įunction f is defined for all real values of x. ![]() It is also important to note that the domain of function f defined above is the set of all the real numbers since f is defined everywhere for all real numbers. What the above says is that if x is smaller than or equal to 2, the formula for the function is f( x ) = -x and if x is greater than 2, the formula is f( x ) = x. Free graph paper is available.Ī piecewise function is usually defined by more than one formula: a fomula for each interval. ![]() The graph, domain and range of these functions and other properties are examined. A step by step tutorial on graphing and sketching piecewise functions.
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