Answer
The domain of this function is all real numbers except $5$ and $2$.
Work Step by Step
To find the domain of this function, we need to find which values are excluded for $t$. In a rational function, the denominator cannot equal $0$ because the function would be undefined. Therefore, we need to set the denominator equal to $0$ and solve for $t$:
$t^2 - 7t + 10 = 0$
We can factor this equation to solve for $t$.
We see that we have a quadratic equation, which is given by the formula:
$ax^2 + bx + c$, where $a$, $b$, and $c$ are all real numbers.
To factor this equation, we want to find which factors when multiplied will give us the product of the $a$ and $c$ terms, which is $10$, but when added together will give us the $b$ term, which is $-7$. This means that both factors should be negative.
Let's look at possible factors:
$-5$ and $-2$
$-10$ and $-1$
It looks like the first combination will work. Let's rewrite the equation in factor form:
$(t - 5)(t - 2) = 0$
According to the zero product property, if the product of two factors equals $0$, then either factor can be $0$; therefore, we can set each of these factors equal to $0$ and solve:
$t - 5 = 0$ or $t - 2 = 0$
Let's look at the first factor:
$t - 5 = 0$
Add $5$ to each side of the equation:
$t = 5$
Let's look at the second factor:
$t - 2 = 0$
Add $2$ to each side:
$t = 2$
By solving for $t$ in the denominator, we find what numbers $t$ cannot be. Therefore, the domain of this function is all real numbers except $5$ and $2$.
