Domain of a Function: Find the domain of the function

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 $\frac{49 + 2x}{x + 4}$, 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 $\frac{23}{5x-2}$, 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, $x\ne\frac{2}{5}$

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 determine the domain of the function $\frac{5+4x}{x^2}$, we need to identify values of $x$ that cause the denominator to be zero, as the function is undefined for these values.

Step 1: Set the denominator equal to zero:

$x^2 = 0$

Step 2: Solve for $x$:

Taking the square root of both sides gives $x = 0$.

The function is undefined at $x = 0$, so we must exclude this value from the domain.

Thus, the domain of the function is all real numbers except $x = 0$.

The domain can be expressed as: $x \ne 0$.

Therefore, the correct answer is option 3: Yes, $x \ne 0$.

Since the denominator is positive for all $x$, the domain of the function is the entire domain.

That is, all values of $x$. Therefore, there is no domain limits.

To determine the domain of the function $\frac{8+8x}{(x+6)^2}$, we must find the values of $x$ that make the function undefined.

This function is a rational function with the numerator $8+8x$ and the denominator $(x+6)^2$. A rational function is undefined where its denominator is equal to zero.

Therefore, we need to solve for $x$ where the denominator equals zero:

• Set the denominator equal to zero: $(x+6)^2 = 0$.
• Take the square root of both sides, resulting in $x+6 = 0$.
• Solve for $x$ by subtracting 6 from both sides: $x = -6$.

This calculation shows that the function is undefined when $x = -6$. Therefore, the domain of the function includes all real numbers except $x = -6$.

Thus, the domain of the function is all real numbers except $x \ne -6$.

The correct choice is:

Yes, $x\ne-6$

The denominator of the function cannot be equal to 0.

Therefore, we will set the denominator equal to 0 and solve for the domain:

$(2x-2)^2\ne0$

$2x\ne2$

$x\ne1$

In other words, the domain of the function is all numbers except 1.