Domain of a Function - Examples, Exercises and Solutions

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.

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

In other words, $x\ne0$

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.

Let's examine the given expression:

$\frac{x}{16}$

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:

$\frac{x}{16}$

the denominator is 16 and:

$16\neq0$

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.

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

Therefore, the domain is all numbers.

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

In other words, the domain is:

$x\ne0$

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

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

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

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

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

This matches choice 3 among the given options.

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

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

$10x - 5 = 0$

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

$10x = 5$

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

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

• Step 5: Simplify the fraction:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• $21x - 7 = 0$

• $21x = 7$

• $x = \frac{7}{21}$

• $x = \frac{1}{3}$

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

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

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

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

To determine the domain of the function $\frac{2x+20}{\sqrt{2x-10}}$, 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, $\sqrt{2x-10}$. For the square root to be valid (for real numbers), we require:

• $2x-10 \geq 0$

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

• Add 10 to both sides: $2x \geq 10$
• Divide both sides by 2: $x \geq 5$

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

• $x > 5$

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

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

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

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

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

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

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

To determine the domain of the function $\frac{49 + 2x}{x + 4}$, 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 $x + 4$.

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

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

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

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

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

Yes, $x \ne -4$

To determine the domain of the function $\frac{5x+2}{4x-3}$, we must identify the values of $x$ 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:

$4x - 3 = 0$

Step 2: Solve for $x$:

$4x = 3$

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

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

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

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

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

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

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

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

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

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

The correct answer choice is:

Yes, $x\ne2$

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

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

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