Content

• Steps
• Method 1 of 6: The Basics
• Method 2 of 6: Domain of Fractional Functions
• Method 3 of 6: Scope of a Rooted Function
• Method 4 of 6: Domain of a Natural Logarithm Function
• Method 5 of 6: Finding a Domain Using a Plot
• Method 6 of 6: Finding a Domain Using a Set

A function domain is a set of numbers on which a function is defined. In other words, these are the values ​​of x that can be substituted into the given equation. The possible values ​​of y are called the range of the function. If you want to find the scope of a function in different situations, follow these steps.

Steps

Method 1 of 6: The Basics

1. 1 Remember what a domain is. The domain of definition is the set of values ​​of x, when substituted into the equation, we get the range of values ​​of y.
2. 2 Learn to find the domain of various functions. The function type determines the method for finding the scope. Here are the basic points that you should know about each type of function, which will be discussed in the next section:
   • Polynomial function without roots or variables in the denominator. For this type of function, the scope is all real numbers.
   • Fractional function with variable in the denominator. To find the domain of a given type of function, equate the denominator to zero and exclude the found values ​​of x.
   • Function with a variable inside the root. To find the scope of a given function type, specify a radical greater than or equal to 0 and find the x values.
   • Natural logarithm function (ln). Enter the expression below the logarithm> 0 and solve.
   • Schedule. Draw a graph to find x.
   • A bunch of. This will be a list of x and y coordinates. The definition area is a list of x coordinates.
3. 3 Correctly designate the area of ​​definition. It is easy to learn how to correctly identify the domain of definition, but it is important that you write down the answer correctly and get high marks. Here are a few things you should know about writing scope:
   • One of the formats for writing the scope: square bracket, 2 end values ​​of the scope, round bracket.
     • For example, [-1; five). This means a range from -1 to 5.
   • Use square brackets [ and ] to indicate that the value is in scope.
     • Thus, in the example [-1; 5) the area includes -1.
   • Use parentheses ( and ) to indicate that the value is not in scope.
     • Thus, in the example [-1; 5) 5 does not belong to the region. The scope only includes values ​​infinitely close to 5, that is, 4.999 (9).
   • Use the U sign to combine areas separated by a gap.
     • For example, [-1; 5) U (5; 10]. This means that the region goes from -1 to 10 inclusive, but does not include 5. This may be for a function where the denominator is "x - 5".
     • You can use multiple Us as needed if the area has multiple gaps / gaps.
   • Use the plus infinity and minus infinity signs to express that the area is infinite in any direction.
     • Always use (), not [] with an infinity sign.

Method 2 of 6: Domain of Fractional Functions

1. 1 Write an example. For example, you are given the following function:
   • f (x) = 2x / (x - 4)
2. 2 For fractional functions with a variable in the denominator, the denominator must be equated to zero. When finding the domain of definition of a fractional function, it is necessary to exclude all values ​​of x at which the denominator is zero, because you cannot divide by zero. Write down the denominator as an equation and set it equal to 0. Here's how to do it:
   • f (x) = 2x / (x - 4)
   • x - 4 = 0
   • (x - 2) (x + 2) = 0
   • x ≠ 2; - 2
3. 3 Write down the scope:
   • x = all real numbers except 2 and -2

Method 3 of 6: Scope of a Rooted Function

1. 1 Write an example. Given a function y = √ (x-7)
2. 2 Set the radical expression to be greater than or equal to 0. You cannot extract the square root of a negative number, although you can extract the square root of 0. Thus, set the radical expression to be greater than or equal to 0. Note that this applies not only to square roots, but also to all roots with an even degree. However, this does not apply to roots with an odd degree, since a negative number can appear under an odd root.
   • x - 7 ≧ 0
3. 3 Highlight the variable. To do this, move 7 to the right side of the inequality:
   • x ≧ 7
4. 4 Write down the scope. There she is:
   • D = [7; + ∞)
5. 5 Find the domain of the rooted function when there are multiple solutions. Given: y = 1 / √ (̅x -4). Setting the denominator to zero and solving this equation will give you x ≠ (2; -2). Here's how you proceed next:
   • Check the area beyond -2 (for example, substituting -3) to make sure that substituting numbers less than -2 in the denominator results in a number greater than 0. And so:
     • (-3) - 4 = 5
   • Now check the area between -2 and +2. Substitute 0 for example.
     • 0 - 4 = -4, so numbers between -2 and 2 don't work.
   • Now try numbers greater than 2, like 3.
     • 3 - 4 = 5, so numbers greater than 2 are fine.
   • Write down the scope. This is how this area is written:
     • D = (-∞; -2) U (2; + ∞)

Method 4 of 6: Domain of a Natural Logarithm Function

1. 1 Write an example. Let's say the function is given:
   • f (x) = ln (x - 8)
2. 2 Specify the expression below the logarithm greater than zero. The natural logarithm must be a positive number, so we set the expression inside the parentheses to be greater than zero.
   • x - 8> 0
3. 3 Decide. To do this, isolate the variable x by adding 8 to both sides of inequality.
   • x - 8 + 8> 0 + 8
   • x> 8
4. 4 Write down the scope. The scope of this function is any number greater than 8. Like this:
   • D = (8; + ∞)

Method 5 of 6: Finding a Domain Using a Plot

1. 1 Take a look at the graph.
2. 2 Check the x values ​​shown on the graph. This may be easier said than done, but here are some tips:
   • Line. If you see a line on the chart that goes to infinity, then all the x values ​​are correct and the scope includes all real numbers.
   • An ordinary parabola. If you see a parabola that looks up or down, then the scope is all real numbers, because all numbers on the x-axis fit.
   • Lying parabola. Now, if you have a parabola with apex at the point (4; 0), which extends infinitely to the right, then the domain D = [4; + ∞)
3. 3 Write down the scope. Write down the scope based on the type of graph you are working with. If you are not sure about the type of graph and you know the function that describes it, plug the x coordinates into the function to test.

Method 6 of 6: Finding a Domain Using a Set

1. 1 Write down the set. A set is a collection of x and y coordinates. For example, you are working with the following coordinates: {(1; 3), (2; 4), (5; 7)}
2. 2 Write down the x coordinates. This is 1; 2; five.
3. 3 Domain: D = {1; 2; five}
4. 4 Make sure set is a function. This requires that every time you substitute the value for x, you get the same value for y. For example, substituting x = 3, you should get y = 6, and so on. The set in the example is not a function, because two different values ​​are given at: {(1; 4), (3; 5), (1; 5)}.