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)=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)=2x6−2x4
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 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 x is not equal to 0 .

(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 (−∞, ∞)
.

Answer 1

To find \((f \cdot g \div h)(x)\), we first need to define what this means:

1. We find \(f(x) \cdot g(x)\).
2. We divide this result by \(h(x)\).

Starting with the functions:

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

Step 1: Calculate \(f(x) \cdot g(x)\)

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

Step 2: Divide by \(h(x)\)

Now we divide this result by \(h(x)\):

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

Step 3: Simplify the expression

We can simplify this:

\[ \frac{x^4 - x^2}{2x^2} = \frac{x^2(x^2 - 1)}{2x^2} = \frac{x^2 - 1}{2}, \quad \text{(for } x \neq 0\text{)} \]

So,

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

Step 4: Determine the domain

The only restriction on the domain comes from \(h(x)\) in the denominator, which cannot be zero. Since:

\[ h(x) = 2x^2 \]

Setting \(h(x) \neq 0\), we have:

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

Thus, the domain is all real numbers except \(x = 0\).

Final Result:

So,

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

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

Therefore, the correct response is:

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