Question

Find the domain of each function. \(P(t)=\frac{\sqrt{t-4}}{3 t-21}\)

Short answer

The domain is \([4, 7) \cup (7, \infty)\).

Step-by-step solution

- Identify the conditions for the square root

For the function to be defined, the expression inside the square root must be non-negative. Therefore, set up the inequality: \[ t - 4 \geq 0 \] Solve for \( t \): \[ t \geq 4 \]

- Determine the conditions for the denominator

The function's denominator cannot be zero, because division by zero is undefined. Set the denominator equal to zero and solve for \( t \): \[ 3t - 21 = 0 \] Solve for \( t \): \[ t = 7 \]

- Combine the conditions

From Step 1, \( t \geq 4 \). From Step 2, \( t eq 7 \). Therefore, combine these conditions to determine the domain: \[ t \geq 4 \], except \( t eq 7 \).

- Write the domain

The domain of \( P(t) \) is: \[ [4, 7) \cup (7, \infty) \]

Key concepts

Inequality Solving

When solving inequalities, the goal is to find the set of values that make the inequality true. In our problem, we encounter an inequality in Step 1: \( t - 4 \geq 0 \). This inequality must be solved to find when the expression inside the square root is non-negative. To solve it, simply rearrange it to get \( t \geq 4 \).

Here's a process you can follow:

• Identify the inequality you need to solve.
• Isolate the variable on one side of the inequality.
• Solve for the variable.
• Write the solution in interval notation.

For instance, solving \( t - 4 \geq 0 \) yields \( t \geq 4 \). This indicates that any value of \( t \) greater than or equal to 4 will satisfy this inequality. It’s crucial when working with inequalities to flip the inequality sign whenever you multiply or divide both sides by a negative number.

But in our case, there’s no need to flip the sign since we are only adding or subtracting terms.

Square Root Conditions

Square roots only accept non-negative values. Therefore, the expression inside a square root must be greater than or equal to zero. In our problem, the expression under the square root is \( t - 4 \).

To determine when the square root is defined, set up the inequality: \( t - 4 \geq 0 \).

Solving this inequality shows that \( t \) must be at least 4. This means the function \( P(t) \) can only be evaluated for values of \( t \) that are 4 or greater.

Remember, the key steps are:

• Identify the expression inside the square root.
• Set up an inequality with the expression being greater than or equal to zero.
• Solve the inequality to find the permissible values.

In our case, \( t \geq 4 \) is the condition, ensuring the square root is defined for non-negative values.

Denominator Restrictions

To avoid undefined expressions, the denominator of any fraction cannot be zero. For our function \( P(t) = \frac{\sqrt{t-4}}{3t-21} \), we must ensure the denominator \( 3t - 21 \) is never zero.

Set up the equation \( 3t - 21 = 0 \) to find the restricted values: \[ 3t - 21 = 0 \] Solving this equation yields: \[ t = 7 \] This means \( t = 7 \) is not allowed because it makes the denominator zero.

To summarize the restrictions, the steps are:

• Write the denominator as an equation set to zero.
• Solve for the variable.
• Exclude the solution from the domain.

In our case, exclude \( t = 7 \) from the domain. This ensures the function is always defined.

Finally, combining the conditions from \( t \geq 4 \) and excluding \( t = 7 \) gives us the domain: \[ [4, 7) \cup (7, \infty) \]