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

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 $x \neq -5$.

To solve this problem, we'll follow these steps to find the domain:

• Step 1: Recognize that the expression $\frac{x+y:3}{2x+6}=4$ involves a fraction. The denominator $2x + 6$ must not be zero, as division by zero is undefined.
• Step 2: Set the denominator equal to zero and solve for $x$ to find the values that must be excluded: $2x + 6 = 0$.
• Step 3: Solve $2x + 6 = 0$:
• $2x + 6 = 0$
• $2x = -6$
• $x = -3$
• Step 4: Conclude that the domain of the function excludes $x = -3$, meaning $x \neq -3$.

Thus, the domain of the given expression is all real numbers except $x = -3$. This translates to:

$x\operatorname{\ne}-3$

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 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, 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.

To find the domain of the expression $\frac{5x+8}{2x-6} = 30$, we need to identify values of $x$ that make the denominator of the fraction zero.

Step 1: Identify the denominator of the fraction, which is $2x - 6$.

Step 2: Set the denominator equal to zero to find the values to exclude:

• Solve the equation $2x - 6 = 0$.
• Add 6 to both sides: $2x = 6$.
• Divide both sides by 2: $x = 3$.

Therefore, $x = 3$ is the value that makes the denominator zero, so it must be excluded from the domain.

Given the choices, the correct answer is $x \neq 3$.

Therefore, the domain of the expression is all real numbers except $x = 3$.

This implies that the correct choice is:

$x \neq 3$

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 find the domain of the function $\frac{2x + 2}{3x - 1}$, we must ensure that the function is defined for all values of $x$ except where the denominator is zero.

Follow these steps to determine the domain:

• Step 1: Identify the denominator of the function. The denominator is $3x - 1$.
• Step 2: Set the denominator equal to zero and solve for $x$: $3x - 1 = 0$
• Step 3: Solve the equation $3x - 1 = 0$: $3x = 1$ $x = \frac{1}{3}$
• Step 4: State the domain. Since the function is undefined at $x = \frac{1}{3}$, the domain consists of all real numbers except $x = \frac{1}{3}$.

The domain of the function $\frac{2x + 2}{3x - 1}$ is therefore all real numbers $x$ such that $x \neq \frac{1}{3}$.

Thus, the solution is: $x \ne \frac{1}{3}$.

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 $x \ne \frac{1}{2}$.

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 $x\ne\frac{3}{5}$.

To solve this problem, we will determine the domain of the rational function by following these steps:

• Step 1: Identify the denominator of the function, which is $9x + 6$.
• Step 2: Set the denominator equal to zero to find values of $x$ that need to be excluded from the domain: $9x + 6 = 0$.
• Step 3: Solve the equation $9x + 6 = 0$ for $x$.
• Step 4: To solve, subtract 6 from both sides to get $9x = -6$.
• Step 5: Divide each side by 9 to solve for $x$, resulting in $x = -\frac{2}{3}$.
• Step 6: The domain of the function excludes the value $x = -\frac{2}{3}$ since it makes the denominator zero.

Thus, the domain of the given function is all real numbers except $x = -\frac{2}{3}$, expressed as $x \ne -\frac{2}{3}$.

Therefore, the correct choice for the domain is: $x\ne-\frac{2}{3}$.

To find the domain of the function $\frac{12}{8x-4}$, we must determine when the denominator equals zero and exclude these values.

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

$8x - 4 = 0$

Step 2: Solve the equation $8x - 4 = 0$ for $x$:

Add 4 to both sides: $8x = 4$

Divide both sides by 8: $x = \frac{4}{8} = \frac{1}{2}$

Step 3: The value $x = \frac{1}{2}$ is where the denominator becomes zero, so this value is excluded from the domain.

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

The domain of the function is $\boxed{x \ne \frac{1}{2}}$.

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 $\frac{24}{21x-7}$, 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 $\frac{5x+2}{4x-3}$, 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:

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

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 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.

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 solve the problem of finding the domain of the function $\frac{5}{2x-\frac{1}{2}}$, we need to ensure the denominator is not zero.

Here is the step-by-step solution:

• Step 1: Identify the denominator of the function, which is $2x - \frac{1}{2}$.
• Step 2: Set the denominator equal to zero: $2x - \frac{1}{2} = 0$.
• Step 3: Solve this equation for $x$:

$2x - \frac{1}{2} = 0$

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

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

Step 4: The calculated value $x = \frac{1}{4}$ is the value that makes the denominator zero. Hence, the domain of the function consists of all real numbers except $x = \frac{1}{4}$.

Therefore, the domain of the function is all real numbers such that $x \neq \frac{1}{4}$.

The correct choice from the multiple choices is: $x \ne \frac{1}{4}$.

The domain of the function is $x \ne \frac{1}{4}$.