Each function contains two types of variables: independent variables and dependent variables whose values literally "depend" on the independent variables. For example, in the function y = f (x) = 2 x + y, x is independent and y is dependent (in other words, y is a function of x). The valid values for a given independent variable x are collectively called the "domain." Valid values for a given dependent variable and are collectively called the "range".
Steps
Part 1 of 3: Finding the Domain of a Function
Step 1. Determine the type of function you are going to work with
The domain of the function is all the values of x (the horizontal axis) that will give you a valid value of y. The equation of the function can be quadratic, a fraction, or contain square roots. To calculate the domain of the function, you must first evaluate the terms within the equation.
- A quadratic function has the form ax2 + bx + c: f (x) = 2 x 2 + 3 x + 4.
- Examples of functions with fractions include: f (x) = (1/ x), f (x) = (x + 1)/(x - 1), etc.
- Functions with a square root include: f (x) = √ x, f (x) = √ (x 2 + 1), f (x) = √- x, etc.
Step 2. Write the domain in the appropriate notation
Writing the domain of a function involves the use of both square brackets "[,]" and parentheses "(,)". You use a bracket when the number is included in the domain and you use parentheses when the domain does not include the number. The letter U indicates a union that connects parts of a domain that could be separated by a space.
- For example, a domain of [-2, 10) U (10, 2] includes -2 and 2 but does not include the number 10.
- Always use parentheses if you are going to use the infinity symbol (∞).
Step 3. Draw a graph of the quadratic equation
Quadratic equations create a parabolic graph that points either up or down. Since the parabola will continue outward infinitely along the x-axis, the domain of most quadratic functions is all real numbers. In other words, a quadratic equation encompasses all the values of x on the number line, which makes its domain R (the symbol for all real numbers).
- To get an idea of the function, choose any value of x and replace it in the function. Solving the function with this value of x will yield a value of y. These x and y values are an (x, y) coordinate of the graph of the function.
- Mark this coordinate and repeat the process with another value of x.
- Labeling a few values in this way should give you a general idea of the form of the quadratic function.
Step 4. Set the denominator to zero if it is a fraction
When working with a fraction, you can never divide by zero. By setting the denominator equal to zero and solving for x, you can calculate the values to be excluded from the function.
- For example: identify the domain of the function f (x) = (x + 1)/(x - 1).
- The denominator of this function is (x - 1).
- Set it equal to zero and solve to find x: x - 1 = 0, x = 1.
- Write the domain: the domain of this function cannot include 1 but includes all real numbers except 1. Therefore, the domain is (-∞, 1) U (1, ∞).
- (-∞, 1) U (1, ∞) can be read as the set of all real numbers excluding 1. The infinity symbol, ∞, represents all real numbers. In this case, all real numbers greater and less than 1 are included in the domain.
Step 5. Set the terms within the root sign as greater than or equal to zero if there is no square root function
You can't take the square root of a negative number; therefore, any value of x that leads to a negative number must be excluded from the domain of that function.
- For example: identify the domain of the function f (x) = √ (x + 3).
- The terms within the root sign are (x + 3).
- Set them as greater than or equal to zero: (x + 3) ≥ 0.
- Solve to find x: x ≥ -3.
- The domain of this function includes all real numbers greater than or equal to -3. Therefore, the domain is [-3, ∞).
Part 2 of 3: Finding the Range of a Quadratic Function
Step 1. Confirm that you have a quadratic function
A quadratic function has the form ax2 + bx + c: f (x) = 2 x 2 + 3 x + 4. The form of a quadratic function on a graph is a parabola that points up or down. There are different methods of calculating the range of a function depending on the type you are working with.
The easiest way to identify the range of other functions, such as square root and fraction functions, is to draw the graph of the function using a graphing calculator
Step 2. Find the x-value of the vertex of the function
The vertex of a quadratic function is the tip of the parabola. Remember: a quadratic function has the form ax2 + bx + c. To find the coordinate of x, use the equation x = -b / 2a. This equation is a derivative of the basic quadratic function that represents the equation with a slope of zero (at the vertex of the graph, the slope of the function is zero).
- For example: find the range of 3 x 2 + 6 x - 2.
- Find the x-coordinate of the vertex: x = -b / 2a = -6 / (2 * 3) = -1.
Step 3. Find the y-value of the vertex of the function
Plug the x-coordinate into the function to find the corresponding y-value of the vertex. This value of y denotes the edge of the range for the function.
- Find the y coordinate: y = 3 x 2 + 6 x - 2 = 3 (-1)2 + 6(-1) -2 = -5.
- The vertex of this function is (-1, -5).
Step 4. Determine the direction of the parabola by substituting in at least one more value for x
Choose any other value of x and replace it in the function to find the corresponding value of y. If the value of y is above the vertex, the parabola continues up to + ∞. If the value of y is below the vertex, the parabola continues until -∞.
- Use the value of x - 2: y = 3 x 2 + 6 x - 2 = y = 3 (-2)2 + 6(-2) – 2 = 12 -12 -2 = -2.
- This produces the coordinate (-2, -2).
- This coordinate tells you that the parabola continues above the vertex (-1, -5). Therefore, the range encompasses all values of and over -5.
- The range of this function is [-5, ∞).
Step 5. Write the range in the appropriate notation
Like the domain, the range is written in the same notation. Use a bracket when the number is included in the domain and a parenthesis when the domain does not include that number. The letter U indicates a union that connects parts of a domain that could be separated by a space.
- For example, a range of [-2, 10) U (10, 2] includes -2 and 2 but does not include the number 10.
- Always use parentheses if you are going to use the infinity symbol (∞).
Part 3 of 3: Graphically Finding the Range of a Function
Step 1. Graph the function
It is often easier to determine the range of a function simply by graphing it. Many square root functions have a range of (-∞, 0] or [0, + ∞) because the vertex of the lateral parabola is on the horizontal axis or the x axis. In this case, the function covers all positive values of y if the parabola goes up or all negative values of y if the parabola goes down. The fraction functions will have asymptotes that define the range.
- Some square root functions will start above or below the x-axis. In this case, the range is determined by the point where the square root function begins. If the parabola starts at y = -4 and goes up, the range is [-4, + ∞).
- The easiest way to graph a function is to use a graphing program or a graphing calculator.
- If you don't have a graphing calculator, you can draw a rough sketch of a graph by plugging in x values into the function and getting the corresponding y values. Plot these coordinates on the graph to get an idea of the shape.
Step 2. Find the minimum of the function
Once you have graphed the function, you should be able to clearly see the lowest point on the graph. If there is no obvious minimum, you should know that some functions will continue until -∞.
A fraction function will include all points except those in the asymptote. They often have ranges like (-∞, 6) U (6, ∞)
Step 3. Determine the maximum of the function
Again, after graphing, you should be able to identify the maximum point of the function. Some functions will continue up to + ∞ and therefore there will not be a maximum.
Step 4. Write the range in the appropriate notation
Like the domain, the range is written in the same notation. Use a square bracket when the number is included in the domain and a parenthesis when the domain does not include that number. The letter U indicates a union that connects parts of a domain that could be separated by a space.
- For example, a range of [-2, 10) U (10, 2] includes -2 and 2 but does not include the number 10.
- Always use parentheses if you are going to use the infinity symbol (∞).