Question

Use the functions below.

f(x)=5x2+3x−6
g(x)=|x+1|−7
h(x)=−x4+2x2+3
What is (f−g−h)(x)? What is the domain of (f−g−h)(x)?

(1 point)
Responses

(f−g−h)(x)=−x4+7x2+3x+|x+1|−10 and the domain of (f−g−h)(x) is (−∞,∞).
open paren f minus g minus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus 3 x plus start absolute value x plus 1 end absolute value minus 10 and the domain of open paren f minus g minus h close paren times x is open paren negative infinity comma infinity close paren .

(f−g−h)(x)=x4+3x2+3x−|x+1|−2 and the domain of (f−g−h)(x) is (−∞,∞).
open paren f minus g minus h close paren times x is equal to x to the 4th power plus 3 x squared plus 3 x minus start absolute value x plus 1 end absolute value minus 2 and the domain of open paren f minus g minus h close paren times x is open paren negative infinity comma infinity close paren .

(f−g−h)(x)=−x4+7x2+3x+|x+1|−10 and the domain of (f−g−h)(x) is x≥−1.
open paren f minus g minus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus 3 x plus start absolute value x plus 1 end absolute value minus 10 and the domain of open paren f minus g minus h close paren times x is x is greater than or equal to negative 1 .

(f−g−h)(x)=x4+3x2+3x−|x+1|−2 and the domain of (f−g−h)(x) is

Answer 1

To find \((f - g - h)(x)\), we need to first express the functions \(f(x)\), \(g(x)\), and \(h(x)\) and then compute the difference.

1. **Functions:**
- \(f(x) = 5x^2 + 3x - 6\)
- \(g(x) = |x + 1| - 7\)
- \(h(x) = -x^4 + 2x^2 + 3\)

2. **Calculation of \((f - g - h)(x)\):**
\[
(f - g - h)(x) = f(x) - g(x) - h(x)
\]

Substituting the functions:
\[
(f - g - h)(x) = (5x^2 + 3x - 6) - (|x + 1| - 7) - (-x^4 + 2x^2 + 3)
\]

Now, simplify this step-by-step:

- Start with \(f(x)\):
\[
5x^2 + 3x - 6
\]

- Subtract \(g(x)\):
\[
5x^2 + 3x - 6 - |x + 1| + 7 = 5x^2 + 3x + 1 - |x + 1|
\]

- Now subtract \(h(x)\):
\[
5x^2 + 3x + 1 - |x + 1| + x^4 - 2x^2 - 3 = x^4 + 3x^2 + 3x - |x + 1| - 2
\]

So we find:
\[
(f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2
\]

3. **Domain of \((f-g-h)(x)\):**
The domain of a function is typically all real numbers unless specified otherwise by pieces like absolute values which can introduce constraints. Here, for \(|x + 1|\), there are no restrictions. Thus:

The domain of \((f - g - h)(x)\) is \((-\infty, \infty)\).

So, the final answer is:
\[
(f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2
\]
with the domain being \((-\infty, \infty)\).