Question

Use the functions below.

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 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 .

(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 (−∞, ∞)
.
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+7x2+4x−7
and the domain of (f+g÷h)(x)
is x≠0
.

Answer 1

To find \((f + \frac{g}{h})(x)\), we need to first compute \(\frac{g(x)}{h(x)}\) and then add \(f(x)\) to that result.

Given:
- \(f(x) = x^4 + 7x^2 - 9\)
- \(g(x) = 8x^2 + 4x\)
- \(h(x) = 2x\)

**Step 1: Compute \(\frac{g(x)}{h(x)}\)**

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

We can simplify this expression:

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

**Step 2: Calculate \((f + \frac{g}{h})(x)\)**

Now, we add \(f(x)\) and \(\frac{g(x)}{h(x)}\):

\[
(f + \frac{g}{h})(x) = f(x) + (4x + 2)
\]

Substituting for \(f(x)\):

\[
(f + \frac{g}{h})(x) = \left(x^4 + 7x^2 - 9\right) + (4x + 2)
\]

Combining like terms:

\[
(f + \frac{g}{h})(x) = x^4 + 7x^2 + 4x - 7
\]

**Step 3: Determine the domain of \((f + \frac{g}{h})(x)\)**

The domain of this expression is determined by the function \(\frac{g(x)}{h(x)}\), specifically by \(h(x)\), since we cannot divide by zero:

\[
h(x) = 2x \Rightarrow h(x) \neq 0 \Rightarrow x \neq 0
\]

Thus, the domain of \((f + \frac{g}{h})(x)\) is:

\[
\text{Domain: } x \neq 0
\]

**Final result:**

\((f + \frac{g}{h})(x) = x^4 + 7x^2 + 4x - 7\) and the domain is \(x \neq 0\).

Based on the provided options in your question, the correct response would be:

\((f+g÷h)(x)=x^4+7x^2+4x−7\) and the domain of \((f+g÷h)(x)\) is \(x \neq 0\).