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

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 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 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 find the domain of the given equation $\frac{xyz}{2(3+y)+4}=8$, we need to ensure the denominator is not zero. This means solving $2(3+y) + 4 = 0$.

Let's solve this step-by-step:

• Step 1: Simplify the expression $2(3+y) + 4 = 0$.
• Step 2: Expand to $6 + 2y + 4 = 0$.
• Step 3: Combine like terms to get $2y + 10 = 0$.
• Step 4: Isolate the variable $y$. Subtract 10 from both sides: $2y = -10$.
• Step 5: Divide by 2 to solve for $y$: $y = -5$.

If $y = -5$, the denominator becomes zero, which makes the original expression undefined.

Therefore, the value of $y$ must not be $-5$ for the expression to be valid. In conclusion, the restriction on $y$ is that $y \neq -5$.

The correct answer choice is: $y \neq -5$.