To determine the smallest value of $a$ such that the function $g(t) = \frac{3t}{5+t^2}$, defined on the interval $(a, \infty)$, has an inverse function, we need to ensure that $g(t)$ is one-to-one on this interval. This can be achieved by ensuring that $g(t)$ is either strictly increasing or strictly decreasing on $(a, \infty)$.

First, we compute the derivative of $g(t)$:

$g'(t) = \frac{d}{dt} \left( \frac{3t}{5 + t^2} \right)$

Using the quotient rule:

$g'(t) = \frac{(5 + t^2) \cdot 3 - 3t \cdot 2t}{(5 + t^2)^2} = \frac{15 + 3t^2 - 6t^2}{(5 + t^2)^2} = \frac{15 - 3t^2}{(5 + t^2)^2}$

For $g(t)$ to be strictly increasing or strictly decreasing, $g'(t)$ should not change sign. This means $\frac{15 - 3t^2}{(5 + t^2)^2}$ must be either always positive or always negative.

Set $g'(t) = 0$ to find the critical points:

$15 - 3t^2 = 0 \implies t^2 = 5 \implies t = \pm \sqrt{5}$

Thus, the function changes concavity at $t = \pm \sqrt{5}$.

To ensure $g(t)$ is one-to-one, we need $g'(t)$ to be either positive or negative on the interval $(a, \infty)$. We can choose the interval such that it does not include points where the derivative changes sign, such as $t = \sqrt{5}$.

Consider the interval $(\sqrt{5}, \infty)$:

• For $t > \sqrt{5}$, $t^2 > 5$, and therefore $15 - 3t^2 < 0$. So $g'(t) < 0$ in $(\sqrt{5}, \infty)$.

Since $g'(t)$ is negative in this interval, $g(t)$ is strictly decreasing on $(\sqrt{5}, \infty)$.

Thus, the smallest value of $a$ such that $g(t)$ has an inverse function is $\sqrt{5}$.

Answer: The smallest value of $a$ is $\sqrt{5}$.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:

1. How do you find the derivative of a function using the quotient rule?
2. What is the importance of a function being one-to-one in having an inverse?
3. How do critical points help determine intervals for monotonicity?
4. Can you provide an example of finding the inverse of a simple function?
5. How do we test if a function is strictly increasing or decreasing?

Tip: When determining if a function has an inverse, always check if it is one-to-one by analyzing its derivative.