Indefinite integral

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 $\frac{1}{5x-4}$, 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, $x\ne\frac{2}{5}$