Use the functions below.

Question

f(x)=x4+7x2−9
g(x)=8x2+4x
h(x)=2x
What is (f+g÷h)(x)
? What is the domain of (f+g÷h)(x)
?

(1 point)
Responses

(f+g÷h)(x)=x4+15x2+6x−9
and the domain of (f+g÷h)(x)
is (−∞, ∞)
.
open paren f plus g divided by h close paren times x is equal to x to the 4th power plus 15 x squared plus 6 x minus 9 and the domain of open paren f plus g divided by h close paren times x is open paren negative infinity comma infinity close paren .

(f+g÷h)(x)=x4+7x2+4x−7
and the domain of (f+g÷h)(x)
is x≠0
.
open paren f plus g divided by h close paren times x is equal to x to the 4th power plus 7 x squared plus 4 x minus 7 and the domain of open paren f plus g divided by h close paren times x is x is not equal to 0 .

(f+g÷h)(x)=x4+7x2+4x−7
and the domain of (f+g÷h)(x)
is (−∞, ∞)
.
open paren f plus g divided by h close paren times x is equal to x to the 4th power plus 7 x squared plus 4 x minus 7 and the domain of open paren f plus g divided by h close paren times x is open paren negative infinity comma infinity close paren .

(f+g÷h)(x)=x4+15x2+6x−9
and the domain of (f+g÷h)(x)
is x≠0
.
open paren f plus g divided by h close paren times x is equal to x to the 4th power plus 15 x squared plus 6 x minus 9 and the domain of open paren f plus g divided by h close paren times x is x is not equal to 0 .

Answer 1

To find \((f+g÷h)(x)\), we need to compute the expression step-by-step:

1. Calculate \(g(x) ÷ h(x)\)
2. Then, substitute that result into \(f(x)\) and add.

Let's find \(g(x) ÷ h(x)\):

• \(g(x) = 8x^2 + 4x\)
• \(h(x) = 2x\)

Thus,

\[ g(x) ÷ h(x) = \frac{g(x)}{h(x)} = \frac{8x^2 + 4x}{2x} \]

We can simplify:

\[ g(x) ÷ h(x) = \frac{8x^2}{2x} + \frac{4x}{2x} = 4x + 2 \quad \text{(for } x \neq 0\text{)} \]

Now, we need to add \(f(x)\) to this result. Recall:

• \(f(x) = x^4 + 7x^2 - 9\)

Now, calculate \((f + g ÷ h)(x)\):

\[ (f + g ÷ h)(x) = f(x) + g(x) ÷ h(x) = (x^4 + 7x^2 - 9) + (4x + 2) \]

Combine the terms:

\[ (f + g ÷ h)(x) = x^4 + 7x^2 + 4x - 9 + 2 = x^4 + 7x^2 + 4x - 7 \]

Now, we check the domain of \((f + g ÷ h)(x)\). The only restriction comes from \(h(x)\) (since we are dividing by it):

• \(h(x) = 2x\) must not be equal to zero, therefore \(x \neq 0\).

Thus, the domain of \((f + g ÷ h)(x)\) is:

\[ \text{Domain: } (-\infty, 0) \cup (0, \infty) \]

Now, based on the calculations, we can conclude: \((f + g ÷ h)(x) = x^4 + 7x^2 + 4x - 7\) and the domain is \(x \neq 0\).

The correct response is: \((f + g ÷ h)(x) = x^4 + 7x^2 + 4x - 7\) and the domain of \((f + g ÷ h)(x)\) is \(x \neq 0\).