Question

Give the intervals on which the given function is continuous. $$ f(t)=\sqrt{5 t^{2}-30} $$

Short answer

The function is continuous on \((-\infty, -\sqrt{6}] \cup [\sqrt{6}, \infty)\).

Step-by-step solution

Determine the Domain of the Square Root Function

The function given is a square root function of the form \( f(t) = \sqrt{5t^2 - 30} \). For the square root to be defined and give real number outputs, the expression inside the square root sign, \(5t^2 - 30\), must be non-negative. So we need to solve the inequality \(5t^2 - 30 \geq 0\).

Solve the Inequality

To solve \(5t^2 - 30 \geq 0\), firstly, simplify it by dividing through by 5 to get \(t^2 - 6 \geq 0\). Next, find the boundary points by setting \(t^2 - 6 = 0\). Solving for \(t\), we get \(t = \pm \sqrt{6}\). Thus, the critical points are \(t = \sqrt{6}\) and \(t = -\sqrt{6}\).

Test Intervals Between Critical Points

Let's test the intervals created by these critical points: \((-\infty, -\sqrt{6})\), \([-\sqrt{6}, \sqrt{6}]\), and \((\sqrt{6}, \infty)\). Select any point within each interval and test it in the inequality \(t^2 - 6 \geq 0\).

Evaluate Each Interval

For \((-\infty, -\sqrt{6})\), choose \(t = -4\). We get \((-4)^2 - 6 = 16 - 6 = 10 \geq 0\), which satisfies the inequality. For \([-\sqrt{6}, \sqrt{6}]\), choose \(t = 0\). We compute \(0^2 - 6 = -6 < 0\), which does not satisfy the inequality. For \((\sqrt{6}, \infty)\), choose \(t = 4\). We calculate \(4^2 - 6 = 16 - 6 = 10 \geq 0\), which satisfies the inequality.

Conclusion on Continuity

Since the function is defined and produces real number outputs for \(t^2 - 6 \geq 0\), the intervals where the function is continuous are \((-\infty, -\sqrt{6}] \cup [\sqrt{6}, \infty)\).

Key concepts

Domain of a Function

When you have a mathematical function, identifying its domain is crucial. The domain of a function refers to all the possible input values, usually denoted by the symbol \( x \) or \( t \) in the function itself, for which the function is defined and outputs real numbers. In the context of the function \( f(t) = \sqrt{5t^2 - 30} \), you first need to focus on what makes the square root valid.

The key rule with square root functions is that the expression inside the root must be non-negative, meaning it should be zero or positive. When we say "non-negative," we mean values that are greater than or equal to zero. For our function, \( 5t^2 - 30 \) has to be \( \geq 0 \) to ensure the square root outputs real numbers. Solving this inequality will give us the interval of \( t \) values that form the domain of the function, ensuring it's valid and continuous.

Always remember:

• The domain contains all values where the function genuinely works (is defined).
• For square root functions, focus on making the inside of the root non-negative.
• Checking the domain helps in analyzing the entire behavior of the function in calculus.

Inequalities in Calculus

In calculus, inequalities play a crucial role as they help you determine where functions are defined or where particular conditions are met. For example, if you encounter an inequality such as \(5t^2 - 30 \geq 0 \), this tells us where the expression inside our square root is valid.

Here's a simple breakdown:

• First, simplify the expression: Divide \(5t^2 - 30 \geq 0 \) by 5, yielding \( t^2 - 6 \geq 0 \).
• Then find critical points by solving \( t^2 - 6 = 0 \). Solving this gives \( t = \pm \sqrt{6} \).
• Break the number line into intervals using these critical points: \(( -\infty, -\sqrt{6})\), \([-\sqrt{6}, \sqrt{6}]\), and \(( \sqrt{6}, \infty)\).
• Test a point in each interval to check if it satisfies the inequality.

These steps help identify which intervals are part of the function's domain, ensuring you work with valid expressions throughout calculations. This practice is not just limited to solving equations but is essential for defining where functions behave as expected across calculus problems.

Square Root Functions

Square root functions are a special type of function involving the square root operation, which presents unique constraints and considerations compared to polynomial or linear functions.

Consider the function \( f(t) = \sqrt{5t^2 - 30} \). To understand a square root function, consider:

• It only takes non-negative numbers. Negative numbers under the root result in complex numbers, outside typical real-valued function considerations.
• Know that anything under the root needs to be zero or positive to maintain the function within the realm of real numbers.
• The behavior of square root functions can often yield two critical points when set equal to zero, which leads to breaking down the function according to its domain constraints.

In general, finding where a square root function is valid and continuous involves assessing the condition of its root expression—making sure it's non-negative to map onto real number solutions. This will define the intervals over which the function can be considered fully continuous and real-valued.