Domain of a Function

What is the Domain of a Function?

The domain of a function includes all those values of X X (independent variable) that, when substituted into the function, keep the function valid and defined.
In simple terms, the domain tells us what numbers we are allowed to plug into the function.

The domain of a function is an integral part of function analysis. Moreover, a definition set is required to create a graphical representation of the function.

How to Find the Domain:

The two most common cases where we encounter restrictions on the domain of a function are:

  1. A variable in the denominator of a fraction: The denominator cannot be zero, as division by zero is undefined.
  2. A variable under a square root or even root: The expression under the root cannot be negative, as square roots of negative numbers are not real numbers.

when we identify one (or more) of the cases, we need to solve it like we usually do, but instead of solving for the solution we'll solve to find the domain:

Variable in the Denominator:
  • Set the denominator not equal to zero denominator0\text{denominator} \neq 0.
  • Solve the resulting equation to find the values to exclude from the domain.

Mathematical function F(X) = 1/X. Explanation of why X ≠ 0 due to division by zero being undefined. Fundamental algebra and function domain restriction concept.

Variable Under a Square Root or Even Root:
  • Set the expression inside the root greater than or equal to zero expression0\text{expression} \geq 0.
  • Solve the inequality to determine the allowed values for the domain.

Mathematical function F(X) = √X. Explanation that a square root cannot be negative, leading to the domain restriction X ≥ 0. Fundamental concept in algebra and function domains.

Although it might seem like most functions don’t have a specific domain, the reality is that every function has a domain. For many functions, the domain is all real numbers, meaning you can plug in any number. However, certain functions, like those with fractions or square roots, have restricted domains. for example, in this function: f(x)=1xf(x) = \frac{1}{x} the domain excludes certain numbers like x=0like~x=0 to avoid breaking mathematical rules.

Suggested Topics to Practice in Advance

  1. Ways to Represent a Function
  2. Representing a Function Verbally and with Tables
  3. Graphical Representation of a Function
  4. Algebraic Representation of a Function
  5. Notation of a Function
  6. Rate of Change of a Function
  7. Variation of a Function
  8. Rate of change represented with steps in the graph of the function
  9. Rate of change of a function represented graphically
  10. Constant Rate of Change
  11. Variable Rate of Change
  12. Rate of Change of a Function Represented by a Table of Values
  13. Functions for Seventh Grade
  14. Increasing and Decreasing Intervals (Functions)
  15. Increasing functions
  16. Decreasing function
  17. Constant Function
  18. Decreasing Interval of a function
  19. Increasing Intervals of a function

Practice Domain of a Function

Examples with solutions for Domain of a Function

Exercise #1

Does the given function have a domain? If so, what is it?

9x4 \frac{9x}{4}

Video Solution

Step-by-Step Solution

Since the function's denominator equals 4, the domain of the function is all real numbers. This means that any one of the x values could be compatible.

Answer

No, the entire domain

Exercise #2

Look at the following function:

5x \frac{5}{x}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

Since the unknown variable is in the denominator, we should remember that the denominator cannot be equal to 0.

In other words, x0 x\ne0

The domain of the function is all those values that, when substituted into the function, will make the function legal and defined.

The domain in this case will be all real numbers that are not equal to 0.

Answer

Yes, x0 x\ne0

Exercise #3

Select the field of application of the following fraction:

x16 \frac{x}{16}

Video Solution

Step-by-Step Solution

Let's examine the given expression:

x16 \frac{x}{16}

As we know, the only restriction that applies to a division operation is division by 0, since no number can be divided into 0 parts, therefore, division by 0 is undefined.

Therefore, when we talk about a fraction, where the dividend (the number being divided) is in the numerator, and the divisor (the number we divide by) is in the denominator, the restriction applies only to the denominator, which must be different from 0,

However in the given expression:

x16 \frac{x}{16}

the denominator is 16 and:

160 16\neq0

Therefore the fraction is well defined and thus the unknown, which is in the numerator, can take any value,

Meaning - the domain (definition range) of the given expression is:

all x

(This means that we can substitute any number for the unknown x and the expression will remain well defined),

Therefore the correct answer is answer B.

Answer

All X All~X

Exercise #4

Select the domain of the following fraction:

8+x5 \frac{8+x}{5}

Video Solution

Step-by-Step Solution

The domain depends on the denominator and we can see that there is no variable in the denominator.

Therefore, the domain is all numbers.

Answer

All numbers

Exercise #5

Select the the domain of the following fraction:

6x \frac{6}{x}

Video Solution

Step-by-Step Solution

The domain of a fraction depends on the denominator.

Since you cannot divide by zero, the denominator of a fraction cannot equal zero.

Therefore, for the fraction 6x \frac{6}{x} , the domain is "All numbers except 0," since the denominator cannot equal zero.

In other words, the domain is:

x0 x\ne0

Answer

All numbers except 0

Exercise #6

Look the following function:

15x4 \frac{1}{5x-4}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 15x4 \frac{1}{5x-4} , we need to find the values of x x for which the function is undefined. This occurs when the denominator equals zero:

First, set the denominator equal to zero:
5x4=0 5x - 4 = 0

Next, solve for x x :
5x=4 5x = 4
x=45 x = \frac{4}{5}

The function is undefined at x=45 x = \frac{4}{5} . Therefore, the domain of the function includes all real numbers except x=45 x = \frac{4}{5} .

In mathematical notation, the domain is:
x45 x \ne \frac{4}{5} .

This matches choice 3 among the given options.

Answer

x45 x\ne\frac{4}{5}

Exercise #7

Look at the following function:

2010x5 \frac{20}{10x-5}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 2010x5 \frac{20}{10x-5} , we need to ensure that the denominator is not zero.

  • Step 1: Identify the denominator, which is 10x5 10x - 5 .
  • Step 2: Set the denominator equal to zero and solve for x x . This gives us the equation:

10x5=0 10x - 5 = 0

  • Step 3: Add 5 to both sides of the equation:

10x=5 10x = 5

  • Step 4: Divide both sides by 10 to isolate x x :

x=510 x = \frac{5}{10}

  • Step 5: Simplify the fraction:

x=12 x = \frac{1}{2}

This means that the function is undefined at x=12 x = \frac{1}{2} . Therefore, the domain of the function is all real numbers except x=12 x = \frac{1}{2} .

Therefore, the domain of the function is x12 x \ne \frac{1}{2} .

Answer

x12 x\ne\frac{1}{2}

Exercise #8

Given the following function:

235x2 \frac{23}{5x-2}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

To determine the domain of the function 235x2 \frac{23}{5x-2} , follow these steps:

  • Step 1: Identify where the function is undefined by setting the denominator equal to zero.
    Equation: 5x2=0 5x - 2 = 0
  • Step 2: Solve the equation for x x .

Let's perform the calculation:
Step 1: Set 5x2=0 5x - 2 = 0 .

Step 2: Solve for x x by adding 2 to both sides:
5x=2 5x = 2

Next, divide both sides by 5:
x=25 x = \frac{2}{5}

This shows that the function is undefined at x=25 x = \frac{2}{5} , thus excluding this point from the domain of the function.

The domain of 235x2 \frac{23}{5x-2} consists of all real numbers except x=25 x = \frac{2}{5} .

Therefore, the domain is expressed as x25 x \ne \frac{2}{5} .

Considering the multiple-choice options, the correct choice is:

Yes, x25 x\ne\frac{2}{5}

Answer

Yes, x25 x\ne\frac{2}{5}

Exercise #9

Given the following function:

2421x7 \frac{24}{21x-7}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 2421x7 \frac{24}{21x-7} , we need to ensure that the denominator is not equal to zero.

Step 1: Set the denominator equal to zero and solve for x x :

  • 21x7=0 21x - 7 = 0

  • 21x=7 21x = 7

  • x=721 x = \frac{7}{21}

  • x=13 x = \frac{1}{3}

The function is undefined when x=13 x = \frac{1}{3} because it would cause division by zero.

Step 2: The domain of the function is all real numbers except x=13 x = \frac{1}{3} .

Therefore, the domain of the function is all x x such that x13 x \neq \frac{1}{3} .

Thus, the correct answer is x13 \boxed{ x\ne\frac{1}{3}} .

Answer

x13 x\ne\frac{1}{3}

Exercise #10

Look at the following function:

2x+202x10 \frac{2x+20}{\sqrt{2x-10}}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 2x+202x10 \frac{2x+20}{\sqrt{2x-10}} , we must ensure that the expression under the square root is non-negative, because the square root of a negative number is not defined in the real numbers.

We start by analyzing the denominator, specifically the square root, 2x10\sqrt{2x-10}. For the square root to be valid (for real numbers), we require:

  • 2x100 2x-10 \geq 0

Now, solve the inequality 2x1002x - 10 \geq 0:

  • Add 10 to both sides: 2x102x \geq 10
  • Divide both sides by 2: x5x \geq 5

However, since the expression 2x102x-10 also prohibits zero in the denominator (as the square root in the denominator cannot be zero), we strictly have:

  • x>5x > 5

Thus, the domain of the function is all xx such that x>5x > 5.

Therefore, the domain of the function 2x+202x10\frac{2x+20}{\sqrt{2x-10}} is x>5 x > 5 .

Answer

x > 5

Exercise #11

Consider the following function:

3x+42x1 \frac{3x+4}{2x-1}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 3x+42x1 \frac{3x+4}{2x-1} , follow these steps:

  • Step 1: Identify the denominator of the rational function, which is 2x1 2x-1 .
  • Step 2: Set the denominator equal to zero to find the values of x x that make the function undefined:
    2x1=0 2x - 1 = 0 .
  • Step 3: Solve for x x :
    Add 1 to both sides: 2x=1 2x = 1 .
    Divide both sides by 2: x=12 x = \frac{1}{2} .

The value x=12 x = \frac{1}{2} makes the denominator zero, which means the function 3x+42x1 \frac{3x+4}{2x-1} is undefined at x=12 x = \frac{1}{2} . Therefore, this value must be excluded from the domain.

The domain of the function is all real numbers except x=12 x = \frac{1}{2} .

Therefore, the solution to the problem is x12 x \ne \frac{1}{2} .

Answer

x12 x\ne\frac{1}{2}

Exercise #12

Given the following function:

49+2xx+4 \frac{49+2x}{x+4}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

To determine the domain of the function 49+2xx+4 \frac{49 + 2x}{x + 4} , we need to focus on avoiding division by zero, which occurs when the denominator is zero.

Let's identify the denominator of the function:

  • The denominator is x+4 x + 4 .

Next, we set the denominator equal to zero and solve for x x :

  • x+4=0 x + 4 = 0
  • Subtract 4 from both sides: x=4 x = -4

This calculation shows that the function is undefined when x=4 x = -4 . Thus, the domain of the function is all real numbers except x=4 x = -4 .

Therefore, the domain of the function is x4 x \neq -4 .

In terms of the provided choices, this corresponds to choice 4:

Yes, x4 x \ne -4

Answer

Yes, x4 x\ne-4

Exercise #13

Look at the following function:

5x+24x3 \frac{5x+2}{4x-3}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 5x+24x3 \frac{5x+2}{4x-3} , we must identify the values of xx that make the denominator zero, as these values are not allowed in the domain of a rational function.

Step 1: Set the denominator equal to zero:

4x3=0 4x - 3 = 0

Step 2: Solve for xx:

4x=3 4x = 3

x=34 x = \frac{3}{4}

The function is undefined at x=34x = \frac{3}{4} because division by zero is not permissible.

Therefore, the domain of the function is all real numbers except x=34x = \frac{3}{4}. This can be expressed as:

x34 x \ne \frac{3}{4}

The correct answer, based on the choices given, is:

x34 x \ne \frac{3}{4}

Answer

x34 x\ne\frac{3}{4}

Exercise #14

Given the following function:

5x2x \frac{5-x}{2-x}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

To determine the domain of the function 5x2x \frac{5-x}{2-x} , we need to identify and exclude any values of x x that make the function undefined. This occurs when the denominator equals zero.

  • Step 1: Set the denominator equal to zero:
    2x=0 2-x = 0
  • Step 2: Solve for x x :
    Adding x x to both sides gives 2=x 2 = x . Hence, x=2 x = 2 .

This means that the function is undefined when x=2 x = 2 . Therefore, the domain of the function consists of all real numbers except x=2 x = 2 .

Thus, the domain is: x2 x \ne 2 .

The correct answer choice is:

Yes, x2 x\ne2

Answer

Yes, x2 x\ne2

Exercise #15

Look at the following function:

10x35x3 \frac{10x-3}{5x-3}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To find the domain of the function 10x35x3\frac{10x-3}{5x-3}, we'll follow these steps:

  • Identify the denominator: B(x)=5x3B(x) = 5x - 3.
  • Set the denominator equal to zero: 5x3=05x - 3 = 0.
  • Solve for xx: Add 3 to both sides, getting 5x=35x = 3. Then, divide by 5: x=35x = \frac{3}{5}.
  • Conclude that the domain is all real numbers except x=35x = \frac{3}{5}, since this makes the denominator zero.

Therefore, the domain of the function is all real numbers except x35 x\ne\frac{3}{5} .

Answer

x35 x\ne\frac{3}{5}