Math Snap

STEP 1

What is this asking?
We're figuring out what inputs we can give to the function $g(t) = \sin(e^t - 3)$ so it produces a valid output!
Watch out!
Don't mix up the rules for sine and exponential functions.
They're both friendly, but in different ways!

STEP 2

1. Domain of the inner function
2. Domain of the outer function
3. Combine for the final domain

STEP 3

Let's define our inner function as $s(t) = e^t - 3$.
We want to find all possible values of $t$ that we can plug into $s(t)$.

STEP 4

The exponential function $e^t$ is defined for all real numbers.
That means $t$ can be anything from negative infinity to positive infinity!
In interval notation, this is $(-\infty, \infty)$.

STEP 5

Subtracting 3 doesn't change what $t$ can be.
We can subtract 3 from any number.
So, the domain of $s(t)$ is still $(-\infty, \infty)$.

STEP 6

Now, let's look at the outer function, which is the sine function.
We're taking the sine of the result of our inner function: $\sin(s(t))$.

STEP 7

The sine function, like the exponential function, is also defined for all real numbers.
No matter what we put inside the sine function, it will happily give us a result.
So, the domain of $\sin(s)$ is $(-\infty, \infty)$.

STEP 8

Since the inner function $s(t)$ produces values in the interval $(-\infty, \infty)$, and the sine function can accept any value in $(-\infty, \infty)$, the function $g(t) = \sin(e^t - 3)$ is defined for all real numbers $t$.

STEP 9

In other words, we can plug any real number into $t$ and get a valid output for $g(t)$!

The domain of $g(t) = \sin(e^t - 3)$ is $(-\infty, \infty)$.