Question

Give the intervals on which the given function is continuous. $$ g(t)=\frac{1}{\sqrt{1-t^{2}}} $$

Short answer

The function \(g(t)\) is continuous on the interval \((-1, 1)\).

Step-by-step solution

Identify the Domain of the Root Function

The function inside the square root is \(1 - t^2\). For the square root to be real and defined, the expression \(1 - t^2\) must be greater than 0. Therefore, \(t^2 < 1\). This inequality suggests that \(t\) must lie between -1 and 1 (not including -1 and 1). The domain of \(\sqrt{1-t^2}\) is \(-1 < t < 1\).

Identify the Domain of the Reciprocal Function

The function \(g(t) = \frac{1}{\sqrt{1-t^2}}\) involves a reciprocal. For this function to be defined, the denominator \(\sqrt{1-t^2}\) cannot be zero. Therefore, \(1-t^2\) must not be zero. Since \(1-t^2 = 0\) when \(t = 1\) or \(t = -1\), the values \(t = 1\) and \(t = -1\) must be excluded from the domain.

Combine the Conditions

Combining the conditions from Step 1 and Step 2, we find that the function \(g(t)\) is continuous over the open interval \((-1, 1)\). Within this interval, \(1-t^2\) is positive, and \(\sqrt{1-t^2}\) is well-defined and non-zero, allowing the reciprocal function to be continuous.

Key concepts

Domain of a Function

The domain of a function is the set of all possible input values (or "x" values in a traditional function context) for which the function is defined. When we talk about the domain, we refer to the values for which the function will produce real and valid outputs.
For a clear understanding, imagine a scenario where you have a simple equation like \[f(x) = x + 3\].Here, any number can be substituted for \(x\), so the domain is all real numbers. But sometimes, functions include limitations, like having square roots or the function being a reciprocal.

For the function \(\sqrt{1-t^2}\), it is crucial that \[1-t^2 > 0\], so the radicand is positive and defined. This implies that the value for \(t\) should not reach -1 or 1.
Hence, the domain for \(\sqrt{1-t^2}\) fits between these values, specifically in the interval from just greater than -1 to just less than 1, written as \((-1, 1)\).
**Note**: Domains are critical because they ensure you are substituting only valid numbers into the function, ensuring it operates correctly.

Reciprocal Functions

Reciprocal functions involve flipping the numerator and denominator of a fraction. A simple example is the reciprocal of 2, which is \[\frac{1}{2}\].Now, for functions:\[g(x) = \frac{1}{f(x)}\].This definition introduces new considerations for the function's domain since the denominator must never be zero.

In our given function \[g(t) = \frac{1}{\sqrt{1-t^2}}\],we need to make sure that \[\sqrt{1-t^2}\]is never zero. This means solving \[1 - t^2 eq 0\],which happens when \(t = 1\) and \(t = -1\). Any attempt to plug these values into the function results in division by zero, which is undefined.
**Key Points**:

• Reciprocals are affected when the denominator is zero.
• Be aware of reciprocal restrictions when determining a function’s domain.
• For \(\frac{1}{\sqrt{1-t^2}}\), the crucial part is ensuring \(1-t^2\) remains strictly positive.

Interval Notation

Interval notation is a method used in mathematics to describe a set of numbers along a particular interval on the number line. This notation provides a concise way to include all the numbers between a given starting and ending point.
For any interval, there are different types depending on whether endpoints are included (closed intervals) or not (open intervals).

In the case of our function \(g(t)\), the interval notation used is \((-1, 1)\). This indicates that all numbers between -1 and 1 are included in the domain, but not -1 and 1 themselves, hence the open brackets.
Here's a brief guide:

• \((a, b)\): Open interval, excluding the endpoints \(a\) and \(b\).
• \([a, b]\): Closed interval, including \(a\) and \(b\).
• \([a, b)\) or \((a, b]\): Semi-open intervals, including one endpoint and excluding the other.

Intervals are essential in accurately representing the sets of numbers where functions behave as intended.
Remember, using precise interval notation is key in mathematics to convey exact ranges and ensure communication is clear and unambiguous.