Domain of a Function: Find the domain of an expression with fractions

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.

To find the domain of the given function $22\left(\frac{2}{x} - 1\right) = 30$, follow these steps:

• Identify critical terms: The term $\frac{2}{x}$ is undefined when $x = 0$ because division by zero is undefined.
• We need to exclude $x = 0$ from the domain to ensure the function remains defined.
• The correct domain for the equation is all real numbers except $x = 0$.

Thus, the domain of the equation is $x \neq 0$.

Therefore, the solution to the problem is $x \neq 0$.

To solve this problem and find the domain for the expression $2x + \frac{6}{x}$, we apply the following steps:

• Step 1: Identify when the fraction $\frac{6}{x}$ is undefined. This occurs when the denominator $x$ equals zero.
• Step 2: To find the restriction, set the denominator equal to zero: $x = 0$.
• Step 3: Solve for $x$ to find the values excluded from the domain. Here, $x \neq 0$.

Since $\frac{6}{x}$ is undefined for $x = 0$, the value $x = 0$ must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.

Therefore, the solution to the problem, indicating the domain of the expression, is $x \neq 0$.

To solve this problem, follow these steps:

• Step 1: Simplify the expression.

Given the function:

$f(x) = \frac{10x}{\frac{1}{2}}$

We can simplify this expression by multiplying by the reciprocal of the denominator:

$f(x) = 10x \times 2 = 20x$

• Step 2: Determine the domain.

Since $f(x) = 20x$ is a linear function, it is defined for all real numbers. There are no restrictions on $x$ since no division by zero or any undefined operations are present.

Conclusion: The domain of the function is all real numbers. This corresponds to choice :

All real numbers

Therefore, the domain of the function is all real numbers.

To solve this problem, we'll follow these steps:

• Step 1: Identify the fraction's denominator.

• Step 2: Determine where this denominator equals zero.

• Step 3: Exclude this value from the domain.

Now, let's work through each step:

Step 1: The given equation is $2x - 3 = \frac{4}{x}$. Notice that the fraction $\frac{4}{x}$ has a denominator of $x$.

Step 2: Set the denominator equal to zero to determine where it is undefined.

$\begin{aligned} x &= 0 \end{aligned}$

Step 3: Since the expression is undefined at $x = 0$, we must exclude this value from the domain.

Therefore, the domain of the expression is all real numbers except 0, formally stated as $x \neq 0$.

The correct solution to the problem is: x ≠ 0.

The task is to find the domain of the function $\frac{23}{x-\frac{1}{4}}$.

Let's consider the denominator $x - \frac{1}{4}$. For the function to be defined, this expression should not be equal to zero, since any number divided by zero is undefined.

• Set the denominator equal to zero:

$x - \frac{1}{4} = 0$

• Solve this equation for $x$:

$x = \frac{1}{4}$

This means that when $x = \frac{1}{4}$, the denominator becomes zero, making the function undefined. Therefore, this $x$ value must be excluded from the domain.

Thus, the domain of the function is all real numbers except $\frac{1}{4}$, which can be represented as:

$x \ne \frac{1}{4}$

This answer matches one of the given choices, specifically choice id="3".

Therefore, the domain of the function $\frac{23}{x-\frac{1}{4}}$ is $x \ne \frac{1}{4}$.

To determine the domain of the function $\frac{3x+4}{\frac{1}{2}x}$, we need to ensure that the denominator does not equal zero, as division by zero is undefined.

First, identify the denominator in the function, which is $\frac{1}{2}x$.

Set the denominator equal to zero to find the values that make it undefined:
$\frac{1}{2}x = 0$

To solve for $x$, multiply both sides by 2 to clear the fraction:
$x = 0$

The domain excludes the value $x = 0$ because it would make the denominator zero, rendering the function undefined.

Thus, the domain of the function $\frac{3x+4}{\frac{1}{2}x}$ is all real numbers except $x = 0$.

Therefore, the correct answer is $x \neq 0$.

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 $\frac{5-x}{2-x}$, 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 determine the domain of the function $\frac{8}{x - 2\frac{1}{2}}$, we'll follow these steps:

• Step 1: Identify where the denominator is zero.

• Step 2: Solve for $x$ in this scenario to find exclusions from the domain.

• Step 3: Provide the domain, excluding these $x$ values.

Let's go through the problem step by step:
Step 1: We note that the function is undefined where the denominator equals zero.
Thus, set the denominator equal to zero: $\begin{aligned} x - 2\frac{1}{2} &= 0 \end{aligned}$
Step 2: Solve for $x$: $\begin{aligned} x &= 2\frac{1}{2} \end{aligned}$
Step 3: The domain of the function is all real numbers except $2\frac{1}{2}$.
Therefore, we express the domain as all real numbers $x$ such that $x \ne 2\frac{1}{2}$.

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

To determine the field of application of the equation $\frac{3x:4}{y+6}=6$, we must identify values of $y$ for which the equation is defined.

• The denominator of the given expression is $y + 6$. In order for the expression to be defined, the denominator cannot be zero.
• This leads us to solve the equation $y + 6 = 0$.
• Solving $y + 6 = 0$ gives us $y = -6$.
• This means $y = -6$ would make the denominator zero, thus the expression would be undefined for this value.

Therefore, the field of application, or the domain of the equation, is all real numbers except $y = -6$.

We must conclude that $y \neq -6$.

Comparing with the provided choices, the correct answer is choice 3: $y \neq -6$.

To find the domain of the function $\frac{4x+4}{x-\frac{1}{8}}$, we need to determine when the denominator equals zero because division by zero is undefined.

Step-by-step approach:

• Step 1: Identify the denominator of the function: $x - \frac{1}{8}$.
• Step 2: Set the denominator equal to zero to find the values of $x$ to exclude: $x - \frac{1}{8} = 0$.
• Step 3: Solve the equation for $x$:
  $x = \frac{1}{8}$.

This means the function is undefined when $x = \frac{1}{8}$. Thus, the domain of the function consists of all real numbers except $x = \frac{1}{8}$.

The domain of the function is therefore all $x$ such that:

$x \ne \frac{1}{8}$

Referring to the answer choices, the correct choice is:

$x\ne\frac{1}{8}$

To solve this problem, we will determine the domain, or field of application, of the equation $\frac{6}{x+5} = 1$.

Step-by-step solution:

• Step 1: Identify the denominator. In the given equation, the denominator is $x+5$.
• Step 2: Determine when the denominator is zero. Solve for $x$ by setting $x+5 = 0$.
• Step 3: Solve the equation: $x+5 = 0$ gives $x = -5$.
• Step 4: Exclude this value from the domain. The domain is all real numbers except $x = -5$.

Therefore, the field of application of the equation is all real numbers except where $x = -5$.

Thus, the domain is $x \neq -5$.

The given function is:

$\frac{5}{5x + 2\frac{1}{2}}$

To determine the domain, we ensure that the denominator is not equal to zero because division by zero is undefined. So, we start by evaluating $5x + 2\frac{1}{2} \neq 0$.
This requires conversion of the mixed number $2\frac{1}{2}$ into an improper fraction or decimal:

$2\frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$

Substituting the improper fraction, our equation becomes:

$5x + \frac{5}{2} \neq 0$

To clear the fraction, multiply through by 2, yielding:

$2 \cdot 5x + 5 \neq 0$

$10x + 5 \neq 0$

Solving for $x$ by subtracting 5 from both sides informs us:

$10x \neq -5$

Now, divide by 10:

$x \neq -\frac{5}{10}$

Simplify the fraction:

$x \neq -\frac{1}{2}$

This solution directly identifies the $x$ value not in the domain:

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

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

Hence, the correct choice from the given options is:

Choice 4: $x \ne -\frac{1}{2}$

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 $\frac{3x+4}{2x-1}$, 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 $x \ne \frac{1}{2}$.