Math / Algebra

QuestionStep 4 (b) $g(t) = \sin(e^t - 3)$ To find the domain of $g(t) = \sin(e^t - 3)$ , we examine the domains of the exponential and sine functions. Remembering that $e^x$ exists for all values of $x$ , the domain $s = e^t - 3$ is what? (Enter your answer using interval notation.) $(-\infty, \infty)$ Step 5 Next, we examine the sine. Since $\sin(x)$ exists for all values of $x$ , then the domain of $y = \sin(s)$ is what? (Enter your answer using interval notation.)

Studdy Solution

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 $g(t)$ !

STEP 10

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

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Studdy Solution

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 $g(t)$ !

STEP 10

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

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Studdy Solution

STEP 1

What is this asking? We're figuring out what inputs we can give to the function g(t)=sin⁡(et−3)g(t) = \sin(e^t - 3)g(t)=sin(et−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 function2. Domain of the outer function3. Combine for the final domain

STEP 3

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

STEP 4

The exponential function ete^tet is defined for *all* real numbers.That means ttt can be anything from negative infinity to positive infinity!In interval notation, this is (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 5

Subtracting **3** doesn't change what ttt can be.We can subtract **3** from any number.So, the domain of s(t)s(t)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))\sin(s(t))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)\sin(s)sin(s) is (−∞,∞)(-\infty, \infty)(−∞,∞).

STEP 8

STEP 9

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

STEP 10

The domain of g(t)=sin⁡(et−3)g(t) = \sin(e^t - 3)g(t)=sin(et−3) is (−∞,∞)(-\infty, \infty)(−∞,∞).

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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!

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

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 $s(t)$ .

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 $(-\infty, \infty)$ .

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

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: $\sin(s(t))$ .

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 $(-\infty, \infty)$ .

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 $(-\infty, \infty)$ .