Respuesta :
Answer:
[tex]\bigg(-2,\displaystyle\frac{-\pi}{2}\bigg) \cup \bigg(\displaystyle\frac{-\pi}{2}, \displaystyle\frac{\pi}{2}\bigg) \cup \bigg(\displaystyle\frac{\pi}{2}, 2\bigg)[/tex]
Step-by-step explanation:
[tex]f(t) = (tan(t) , 8t , ln(4-t^2))[/tex]
In order to find domain of f(t), we have to find individual domain
Domain of tan(t):
t can take values [tex]\frac{\pi}{2} + \pi n, \text{ for n} = ...-2,-1,0,1,2...[/tex]
Domain for 8t:
t can take values [tex](-\infty, \infty)[/tex]
Domain for [tex]ln(4t - t^2)[/tex]:
[tex]4t - t^2 > 0\\t^2 < 4\\-2 < t < 2[/tex]
Now, domain for f(t) is the common for all the three domains.
So domain of f(t) is given by:
[tex]\bigg(-2,\displaystyle\frac{-\pi}{2}\bigg) \cup \bigg(\displaystyle\frac{-\pi}{2}, \displaystyle\frac{\pi}{2}\bigg) \cup \bigg(\displaystyle\frac{\pi}{2}, 2\bigg)[/tex]
For a given function f(x), we define the domain as the set of possible inputs.
We will find that the domain is:
D = { t | -2 ≤ t ≤ 2 | t ≠ -π/2, π/2}
Here we have the function:
f(t) = < tan(t), 8*t, Ln(4 - t^2) >
The domain of this function will be the smaller of the domains of the 3 functions that define f(x).
So we need to find each one of these:
Tan(t)
The domain of the tangent function is the set of all real values, except these values of t that make the cosine function equal to zero, so we write the domain as:
D = { t | t ≠ ... -π/2, π/2, 3π/2, ...}
8*t
This is a linear function, the domain is the set of all real numbers.
ln(4 - t^2)
In the natural logarithm, we can only use inputs equal to or larger than zero.
Then we need to solve:
4 - t^2 ≥ 0
t^2 ≤ 4
-2 ≤ t ≤ 2
Then the domain of f(x) will be:
D = { t | -2 ≤ t ≤ 2 | t ≠ -π/2, π/2}
If you want to learn more, you can read:
https://brainly.com/question/15339465
