Use the functions below.

Question

f(x)=x2−1
g(x)=x2
h(x)=2x2
What is (f⋅g÷h)(x)
? What is the domain of (f⋅g÷h)(x)
?

(1 point)
Responses

(f⋅g÷h)(x)=2x6−2x4
and the domain of (f⋅g÷h)(x)
is (−∞, ∞)
.
open paren f times g divided by h close paren times x is equal to 2 x to the 6th power minus 2 x to the 4th power and the domain of open paren f times g divided by h close paren times x is open paren negative infinity comma infinity close paren .

(f⋅g÷h)(x)=12x2−12
and the domain of (f⋅g÷h)(x)
is x≠0
.
open paren f times g divided by h close paren times x is equal to 1 half x squared minus 1 half and the domain of open paren f times g divided by h close paren times x is x is not equal to 0 .

(f⋅g÷h)(x)=12x2−12
and the domain of (f⋅g÷h)(x)
is (−∞, ∞)
.
open paren f times g divided by h close paren times x is equal to 1 half x squared minus 1 half and the domain of open paren f times g divided by h close paren times x is open paren negative infinity comma infinity close paren .

(f⋅g÷h)(x)=2x6−2x4
and the domain of (f⋅g÷h)(x)
is x≠0
.

Answer 1

To find \((f \cdot g \div h)(x)\), we first need to calculate \(f(x)\), \(g(x)\), and \(h(x)\):

1. \(f(x) = x^2 - 1\)
2. \(g(x) = x^2\)
3. \(h(x) = 2x^2\)

Now, we can calculate \(f(x) \cdot g(x)\):

\[ f(x) \cdot g(x) = (x^2 - 1) \cdot x^2 = x^4 - x^2 \]

Next, we will divide this result by \(h(x)\):

\[ f \cdot g \div h = \frac{x^4 - x^2}{2x^2} \]

Now, we can simplify this expression:

\[ = \frac{x^2(x^2 - 1)}{2x^2} = \frac{x^2 - 1}{2} \quad \text{(where } x \neq 0 \text{)} \]

Thus, we have:

\[ (f \cdot g \div h)(x) = \frac{1}{2} (x^2 - 1) = \frac{1}{2} x^2 - \frac{1}{2} \]

Now, let's determine the domain of \((f \cdot g \div h)(x)\). The only restriction comes from \(h(x)\) in the denominator, where \(h(x) = 2x^2\), which cannot be zero. Therefore, we need:

\[ 2x^2 \neq 0 \implies x^2 \neq 0 \implies x \neq 0 \]

Putting this all together, we have:

\[ (f \cdot g \div h)(x) = \frac{1}{2} x^2 - \frac{1}{2} \]

The domain of \((f \cdot g \div h)(x)\) is \(x \neq 0\).

So, the answer is:

\((f \cdot g \div h)(x) = \frac{1}{2} x^2 - \frac{1}{2}\) and the domain is \(x \neq 0\).