Question

Question
Use the functions below.

f(x)=−5x2+4
g(x)=−4x2−7x+9
h(x)=2x2+x−2
What is (f−g−h)(x)
? What is the domain of (f−g−h)(x)
?

(1 point)
Responses

(f−g−h)(x)=−7x2−6x+11
and the domain of (f−g−h)(x)
is x≥4
.
open paren f minus g minus h close paren times x is equal to negative 7 x squared minus 6 x plus 11 and the domain of open paren f minus g minus h close paren times x is x is greater than or equal to 4 .

(f−g−h)(x)=−7x2−6x+11
and the domain of (f−g−h)(x)
is (−∞,∞)
.
open paren f minus g minus h close paren times x is equal to negative 7 x squared minus 6 x plus 11 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)=−3x2+6x−3
and the domain of (f−g−h)(x)
is x≥4
.
open paren f minus g minus h close paren times x is equal to negative 3 x squared plus 6 x minus 3 and the domain of open paren f minus g minus h close paren times x is x is greater than or equal to 4 .

(f−g−h)(x)=−3x2+6x−3
and the domain of (f−g−h)(x)
is (−∞,∞)
.

Answer 1

To find \((f-g-h)(x)\), we start by calculating \(f(x) - g(x) - h(x)\).

1. **Given functions**:
- \(f(x) = -5x^2 + 4\)
- \(g(x) = -4x^2 - 7x + 9\)
- \(h(x) = 2x^2 + x - 2\)

2. **Calculating \(f - g\)**:
\[
f(x) - g(x) = (-5x^2 + 4) - (-4x^2 - 7x + 9)
\]
Simplifying this:
\[
= -5x^2 + 4 + 4x^2 + 7x - 9
\]
\[
= (-5x^2 + 4x^2) + 7x + (4 - 9)
\]
\[
= -x^2 + 7x - 5
\]

3. **Now calculate \((f - g) - h\)**:
\[
(f - g)(x) - h(x) = (-x^2 + 7x - 5) - (2x^2 + x - 2)
\]
Simplifying this:
\[
= -x^2 + 7x - 5 - 2x^2 - x + 2
\]
\[
= (-x^2 - 2x^2) + (7x - x) + (-5 + 2)
\]
\[
= -3x^2 + 6x - 3
\]

Thus, we find:
\[
(f - g - h)(x) = -3x^2 + 6x - 3
\]

4. **Finding the domain**:
The functions \(f(x)\), \(g(x)\), and \(h(x)\) are all polynomials and thus are defined for all real numbers. Therefore, the domain of \((f - g - h)(x)\) is \((-\infty, \infty)\).

Final answer:
\[
(f - g - h)(x) = -3x^2 + 6x - 3 \quad \text{and the domain of } (f - g - h)(x) \text{ is } (-\infty, \infty).
\]

The correct option is:
\((f-g-h)(x)=-3x^2+6x-3\) and the domain of \((f-g-h)(x)\) is \((-∞,∞)\).