Content

• Steps
• Part 1 of 3: Finding the Domain of a Function
• Part 2 of 3: Finding the Range of a Quadratic Function
• Part 3 of 3: Finding the Range of a Function Using Its Graph

Each function has two variables - the independent variable and the dependent variable, whose values ​​depend on the values ​​of the independent variable. For example, in the function y = f(x) = 2x + y the independent variable is x and the dependent variable is y (in other words, y is a function of x). The valid values ​​of the independent variable "x" are called the domain of the function, and the valid values ​​of the dependent variable "y" are called the domain of the function.

Steps

Part 1 of 3: Finding the Domain of a Function

1. 1 Determine the type of function given to you. The range of values ​​of the function are all admissible values ​​of "x" (plotted along the horizontal axis), which correspond to the admissible values ​​of "y". The function can be quadratic or contain fractions or roots. To find the domain of a function, you first need to determine the type of the function.
   • The quadratic function is: ax + bx + c: f (x) = 2x + 3x + 4
   • A function containing a fraction: f (x) = (/_x), f (x) = /_{(x - 1)} (etc).
   • Root-containing function: f (x) = √x, f (x) = √ (x + 1), f (x) = √-x (and so on).
2. 2 Select the appropriate entry for the scope of the function. The scope is written in square and / or parentheses. A square bracket is used when a value is within the scope of a function; if the value is not in scope, a parenthesis is used. If the function has several non-contiguous domains of definition, the symbol "U" is placed between them.
   • For example, the domain [-2,10) U (10,2] includes the values ​​-2 and 2, but does not include the value 10.
   • Parentheses are always used with the infinity symbol ∞.
3. 3 Plot a quadratic function. The graph of such a function is a parabola, the branches of which are directed either up or down. Since the parabola increases or decreases on the entire X-axis, the domain of the quadratic function is all real numbers. In other words, the domain of such a function is the set R (R denotes all real numbers).
   • For a better understanding of the concept of a function, choose any value of "x", substitute it into the function and find the value "y". The pair of values ​​"x" and "y" represent a point with coordinates (x, y), which lies on the graph of the function.
   • Draw this point on the coordinate plane and follow the described process with a different "x" value.
   • By plotting several points on the coordinate plane, you will get a general idea of ​​the shape of the function graph.
4. 4 If the function contains a fraction, set its denominator to zero. Remember that you cannot divide by zero. Therefore, equating the denominator to zero, you will find values ​​for "x" that are not in the scope of the function.
   • For example, find the domain of the function f (x) = /_{(x - 1)}.
   • Here the denominator is (x - 1).
   • Equate the denominator to zero and find "x": x - 1 = 0; x = 1.
   • Write down the scope of the function. The domain does not include 1, that is, it includes all real numbers except 1. Thus, the domain of the function is: (-∞, 1) U (1, ∞).
   • The notation (-∞, 1) U (1, ∞) reads like this: the set of all real numbers except 1. The infinity symbol ∞ means all real numbers. In our example, all real numbers greater than 1 and less than 1 are included in the scope.
5. 5 If the function contains a square root, then the radical expression must be greater than or equal to zero. Remember that the square root of negative numbers is not extracted. Therefore, any value of "x" at which the radical expression becomes negative must be excluded from the scope of the function.
   • For example, find the domain of the function f (x) = √ (x + 3).
   • The radical expression: (x + 3).
   • The radical expression must be greater than or equal to zero: (x + 3) ≥ 0.
   • Find "x": x ≥ -3.
   • The scope of this function includes the set of all real numbers that are greater than or equal to -3. Thus, the domain is [-3, ∞).

Part 2 of 3: Finding the Range of a Quadratic Function

1. 1 Make sure you are given a quadratic function. The quadratic function has the form: ax + bx + c: f (x) = 2x + 3x + 4. The graph of such a function is a parabola whose branches are directed either up or down. There are various methods for finding the range of values ​​of a quadratic function.
   • The easiest way to find the range of a root or fraction function is to graph that function using a graphing calculator.
2. 2 Find the x-coordinate of the vertex of the function graph. In the case of a quadratic function, find the x-coordinate of the vertex of the parabola. Remember that the quadratic function is: ax + bx + c. To calculate the x-coordinate, use the following equation: x = -b / 2a. This equation is a derivative of the fundamental quadratic function and describes a tangent, the slope of which is zero (the tangent to the vertex of the parabola is parallel to the X axis).
   • For example, find the range of the function 3x + 6x -2.
   • Calculate the x-coordinate of the vertex of the parabola: x = -b / 2a = -6 / (2 * 3) = -1
3. 3 Find the y-coordinate of the vertex of the function graph. To do this, substitute the found coordinate "x" into the function. The sought-for coordinate "y" is the limiting value of the range of values ​​of the function.
   • Calculate the y-coordinate: y = 3x + 6x - 2 = 3 (-1) + 6 (-1) -2 = -5
   • The coordinates of the vertex of the parabola of this function are (-1, -5).
4. 4 Determine the direction of the parabola by substituting at least one x value into the function. Pick any other x value and plug it into the function to calculate the corresponding y value. If the found value "y" is greater than the coordinate "y" of the vertex of the parabola, then the parabola is directed upwards. If the found value "y" is less than the coordinate "y" of the vertex of the parabola, then the parabola is directed downward.
   • Substitute x = -2 in the function: y = 3x + 6x - 2 = y = 3 (-2) + 6 (-2) - 2 = 12 -12 -2 = -2.
   • The coordinates of the point on the parabola are (-2, -2).
   • The coordinates found indicate that the branches of the parabola are directed upward. Thus, the function range includes all y values ​​that are greater than or equal to -5.
   • Range of values ​​of this function: [-5, ∞)
5. 5 The range of values ​​of a function is written in the same way as the range of definition of a function. The square bracket is used when the value is in the range of the function; if the value is not in the range, a parenthesis is used. If the function has several non-contiguous ranges of values, the symbol "U" is placed between them.
   • For example, the range [-2,10) U (10,2] includes the values ​​-2 and 2, but does not include the value 10.
   • Parentheses are always used with the infinity symbol ∞.

Part 3 of 3: Finding the Range of a Function Using Its Graph

1. 1 Plot the function. In many cases, it is easier to find the range of values ​​of a function by plotting its graph. The range of values ​​of many functions with roots is (-∞, 0] or [0, + ∞), since the vertex of the parabola directed to the right or to the left lies on the X-axis. In this case, the range includes all positive values ​​of "y" if the parabola is increasing, or all negative y values ​​if the parabola is decreasing. Fractional functions have asymptotes that define their range.
   • The vertices of the graphs of some functions with roots lie above or below the X-axis. In this case, the range of values ​​is determined by the “y” coordinate of the parabola vertex. If, for example, the coordinate "y" of the vertex of a parabola is -4 (y = -4), and the parabola is increasing, then the range of values ​​is [-4, + ∞).
   • The easiest way to graph a function is to use a graphing calculator or special software.
   • If you don't have a graphing calculator, create a rough graph by plugging in multiple x values ​​into the function and calculating the corresponding y values. Plot the found points on the coordinate plane to get a general idea of ​​the shape of the graph.
2. 2 Find the minimum of the function. Once you have plotted a function, you will see the point at which the function has a minimum value.If there is no obvious minimum, then it does not exist, and the graph of the function goes to -∞.
   • The range of values ​​of the function includes all values ​​of "y" except for the values ​​of the asymptotes. Often, the ranges of values ​​of such functions are written as follows: (-∞, 6) U (6, ∞).
3. 3 Determine the maximum of the function. Once you have plotted a function, you will see the point at which the function has its maximum value. If there is no visual maximum, then it does not exist, and the graph of the function goes to + ∞.
4. 4 The range of values ​​of a function is written in the same way as the range of definition of a function. The square bracket is used when the value is in the range of the function; if the value is not in the range, a parenthesis is used. If the function has several non-contiguous ranges of values, the symbol "U" is placed between them.
   • For example, the range [-2,10) U (10,2] includes the values ​​-2 and 2, but does not include the value 10.
   • Parentheses are always used with the infinity symbol ∞.