Each function contains two types of variables: independent and dependent ones, the value of the latter literally "depends" on that of the former. For example, in the function y = f (x) = 2 x + y, x is the independent variable and y is dependent (in other words, y is a function of x). The set of valid values that are assigned to the independent variable x is called the "domain". The set of valid values assumed by the dependent variable y is called "range".
Steps
Part 1 of 3: Finding the Domain of a Function
Step 1. Determine the type of function under consideration
The domain of a function is represented by all the values of x (arranged on the abscissa axis) which make the variable y assume a valid value. The function could be quadratic, a fraction, or contain roots. To calculate the domain of a function, you must first evaluate the terms it contains.
- A second degree equation respects the form: ax2 + bx + c. For example: f (x) = 2x2 + 3x + 4.
- Functions with fractions include: f (x) = (1/x), f (x) = (x + 1)/(x - 1) and so on.
- Equations with a root look like this: f (x) = √x, f (x) = √ (x2 + 1), f (x) = √-x and so on.
Step 2. Write the domain respecting the correct notation
To define the domain of a function you must use both square brackets [,] and round brackets (,). You use the square ones when the extreme of the set is included in the domain, while you must opt for the round ones if the extreme of the set is not included. The capital letter U indicates the union between two parts of the domain that can be separated by a portion of values excluded from the domain.
- For example, the domain [-2, 10) U (10, 2] includes the values of -2 and 2, but excludes the number 10.
- Always use round brackets when you need to use the infinity symbol, ∞.
Step 3. Plot the second degree equation
This type of function generates a parabola that can be pointing up or down. This parabola continues its extension to infinity, well beyond the abscissa axis that you have drawn. The domain of most quadratic functions is the set of all real numbers. In other words, a second degree equation includes all the values of x represented on the number line, consequently its domain is R. (the symbol that indicates the set of all real numbers).
- To determine the type of function under consideration, assign any value to x and insert it into the equation. Solve it based on the chosen value and find the corresponding number for y. The pair of x and y values represent the (x; y) coordinates of a point on the function graph.
- Locate the point with these coordinates and repeat the process for another x value.
- If you draw some points obtained with this method on the Cartesian axis system, you can get a rough idea of the shape of the quadratic function.
Step 4. Set the denominator to zero if the function is a fraction
When working with a fraction, you can never divide the numerator by zero. If you set the denominator to zero and solve the equation for x, you find the values that should be excluded from the function.
- For example, suppose we need to find the domain of f (x) = (x + 1)/(x - 1).
- The denominator of the function is (x - 1).
- Set the denominator to zero and solve the equation for x: x - 1 = 0, x = 1.
- At this point, you can write the domain which cannot include the value 1 but all real numbers except 1. So the domain written in the correct notation is: (-∞, 1) U (1, ∞).
- The notation (-∞, 1) U (1, ∞) can be read as: all real numbers except 1. The infinity symbol (∞) represents all real numbers. In this case, all those greater and less than 1 are part of the domain.
Step 5. Set the terms within the square root as zero or greater if you are working with an equation of roots
Since you cannot take the square root of a negative number, you must exclude from the domain all values of x that lead to a radicand less than zero.
- For example, identify the domain of f (x) = √ (x + 3).
- The rooting is (x + 3).
- Make this zero or greater: (x + 3) ≥ 0.
- Solve the inequality for x: x ≥ -3.
- The domain of the function is represented by all real numbers greater than or equal to -3, therefore: [-3, ∞).
Part 2 of 3: Finding the Codomain of a Quadratic Function
Step 1. Make sure it's a quadratic function
This type of equation respects the form: ax2 + bx + c, for example f (x) = 2x2 + 3x + 4. The graphical representation of a quadratic function is a parabola pointing up or down. There are several methods to calculate the range of a function based on which typology it belongs to.
The easiest way to find the range of other functions, such as fractional or rooted ones, is to graph them with a scientific calculator
Step 2. Find the value of x at the vertex of the function
The vertex of a second degree function is the "tip" of the parabola. Remember that this kind of equation respects the form: ax2 + bx + c. To find the coordinate on the abscissas use the equation x = -b / 2a. This equation is a derivative of the basic quadratic function with slope equal to zero (at the vertex of the graph the slope of the function - or angular coefficient - is zero).
- For example, find the range of 3x2 + 6x -2.
- Calculate the coordinate of x at the vertex x = -b / 2a = -6 / (2 * 3) = -1;
Step 3. Calculate the value of y at the vertex of the function
Enter the value of the ordinates at the vertex in the function and find the corresponding number of ordinates. The result indicates the end of the range of the function.
- Calculate the coordinate of y: y = 3x2 + 6x - 2 = 3 (-1)2 + 6(-1) -2 = -5.
- The vertex coordinates of this function are (-1; -5).
Step 4. Determine the direction of the parabola by inserting at least one other value for x into the equation
Choose another number to assign to the abscissa and calculate the corresponding ordinate. If the value of y is above the vertex, then the parabola continues towards + ∞. If the value is below the vertex, the parabola extends to -∞.
- Make x the value of -2: y = 3x2 + 6x - 2 = y = 3 (-2)2 + 6(-2) – 2 = 12 -12 -2 = -2.
- From the calculations you get the pair of coordinates (-2; -2).
- This pair makes you understand that the parabola continues above the vertex (-1; -5); therefore the range includes all y values greater than -5.
- The range of this function is [-5, ∞).
Step 5. Write the range with the correct notation
This is identical to the one used for the domain. Use square brackets when the extreme is included in the range and round brackets to exclude it. The capital letter U indicates the union between two parts of the range that are separated by a portion of values not included.
- For example, the range of [-2, 10) U (10, 2] includes the values -2 and 2, but excludes 10.
- Always use round brackets when considering the infinity symbol, ∞.
Part 3 of 3: Graphically Finding the Range of a Function
Step 1. Draw the graph
Often the easiest way to find the range of a function is to graph it. Many functions with roots have a range of (-∞, 0] or [0, + ∞) because the vertex of the horizontal parabola is on the abscissa axis. In this case, the function includes all positive values of y, if the half-parabola goes up, and all negative values, if the half-parabola goes down. Functions with fractions have asymptotes that define the range.
- Some functions with radicals have a graph that originates above or below the abscissa axis. In this case, the range is determined by where the function starts. If the parabola originates in y = -4 and tends to rise, then its range is [-4, + ∞).
- The simplest way to graph a function is to use a scientific calculator or a dedicated program.
- If you don't have such a calculator, you can sketch on paper by entering values for x into the function and calculating the correspondents for y. Find on the graph the points with the coordinates you calculated, to get an idea of the shape of the curve.
Step 2. Find the minimum of the function
When you have drawn the graph, you should be able to clearly identify the minus point. If there is no well-defined minimum, know that some functions tend to -∞.
A function with fractions will include all points except those found on the asymptote. In this case, the range takes values such as (-∞, 6) U (6, ∞)
Step 3. Find the maximum of the function
Again, the graphical representation is of great help. However, some functions tend to + ∞ and, consequently, do not have a maximum.
Step 4. Write the range respecting the right notation
Just like with the domain, the range must also be expressed with square brackets when the extreme is included and with rounds when the extreme value is excluded. The capital letter U indicates the union between two parts of the range that are separated by a portion that is not part of it.
- For example, the range [-2, 10) U (10, 2] includes the values of -2 and 2, but excludes 10.
- When using the infinity symbol, ∞, always use round brackets.
