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 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 $\frac{1}{5x-4}$, 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 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 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 $\frac{10x-3}{5x-3}$, 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 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 $\frac{20}{10x-5}$, 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 $\frac{5x+2}{4x-3}$, 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 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, 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 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 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 solve for the domain of the function $\frac{65}{(2x-2)^2}$, follow these steps:

• Step 1: Identify when the denominator $(2x-2)^2$ equals zero because the denominator cannot be zero.
• Step 2: Solve the equation $(2x-2)^2 = 0$. This simplifies to $2x-2 = 0$.
• Step 3: Solve for $x$ by adding 2 to both sides: $2x = 2$.
• Step 4: Divide both sides by 2 to isolate $x$: $x = 1$.
• Step 5: The value $x = 1$ makes the denominator zero, so it must be excluded from the domain.

Thus, the domain of the function is all real numbers except $x \ne 1$.

Therefore, the solution to the problem is $x \ne 1$.

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{align*} x - 2\frac{1}{2} &= 0 \end{align*} Step 2: Solve for $x$: \begin{align*} x &= 2\frac{1}{2} \end{align*} 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 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 find the domain of the function $\frac{5x+15}{10x+\frac{1}{2}}$, we must ensure the denominator is not zero.

The critical expression to consider is the denominator:

$10x + \frac{1}{2}$

Let's solve the equation:

1. Set the denominator equal to zero: $10x + \frac{1}{2} = 0$.
2. To clear the fraction, multiply everything by 2: $2(10x) + 2\left(\frac{1}{2}\right) = 0$.
3. This simplifies to: $20x + 1 = 0$.
4. Subtract 1 from both sides: $20x = -1$.
5. Divide by 20: $x = -\frac{1}{20}$.

Thus, the function is undefined when $x = -\frac{1}{20}$. Consequently, the domain of the function is all real numbers except $x = -\frac{1}{20}$.

Therefore, the solution to the problem is $x \ne -\frac{1}{20}$.