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+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+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 x≥−1
.

Answer 1

To find \((f - g - h)(x)\), we first need to consider each function:

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

Now let's compute \((f - g - h)(x)\):

\[
(f - g - h)(x) = f(x) - g(x) - h(x)
\]

Substituting the expressions for \(f(x)\), \(g(x)\), and \(h(x)\):

\[
= (5x^2 + 3x - 6) - (|x + 1| - 7) - (-x^4 + 2x^2 + 3)
\]

Now let's simplify this step by step:

1. Rewrite \(g(x)\) as \(-|x + 1| + 7\):
\[
-g(x) = -|x + 1| + 7
\]

2. Substitute \(h(x)\) as \(+x^4 - 2x^2 - 3\):
\[
-h(x) = x^4 - 2x^2 - 3
\]

3. Combining everything:
\[
(f - g - h)(x) = (5x^2 + 3x - 6) + (-|x + 1| + 7) + (x^4 - 2x^2 - 3)
\]

Now collect the like terms:

- The \(x^4\) term: \(x^4\)
- The \(x^2\) terms: \(5x^2 - 2x^2 = 3x^2\)
- The \(x\) terms: \(3x\)
- The constant terms: \(-6 + 7 - 3 = -2\)

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

Therefore, the final expression for \( (f−g−h)(x) \) is:
\[
(f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2
\]

Next, we determine the domain of \((f - g - h)(x)\). Since \(f(x)\), \(g(x)\), and \(h(x)\) are all polynomials and the absolute value function, they are defined for all real numbers. Thus:

The domain of \((f - g - h)(x)\) is:
\[
(-\infty, \infty)
\]

In summary:
\[
(f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2 \quad \text{and the domain is} \quad (-\infty, \infty).
\]