Question

Use the functions below.

f(x)=x5−2x4+5x−11
g(x)=9x−1
h(x)=x−−√3+6
What is (f+g+h)(x)
? What is the domain of (f+g+h)(x)
?

(1 point)
Responses

(f+g+h)(x)=x5−2x4+14x+x−−√3−6
and the domain of (f+g+h)(x)
is (−∞, ∞)
.
open paren f plus g plus h close paren times x is equal to x to the 5th power minus 2 x to the 4th power plus 14 x plus cube root of x minus 6 and the domain of open paren f plus g plus h close paren times x is open paren negative infinity comma infinity close paren .

(f+g+h)(x)=x5−2x4+14x+x−−√3−6
and the domain of (f+g+h)(x)
is x≥0
.
open paren f plus g plus h close paren times x is equal to x to the 5th power minus 2 x to the 4th power plus 14 x plus cube root of x minus 6 and the domain of open paren f plus g plus h close paren times x is x is greater than or equal to 0 .

(f+g+h)(x)=10x5−3x4+5x+x−−√3−6
and the domain of (f+g+h)(x)
is (−∞, ∞)
.
open paren f plus g plus h close paren times x is equal to 10 x to the 5th power minus 3 x to the 4th power plus 5 x plus cube root of x minus 6 and the domain of open paren f plus g plus h close paren times x is open paren negative infinity comma infinity close paren .

(f+g+h)(x)=10x5−3x4+5x+x−−√3−6
and the domain of (f+g+h)(x)
is x≥0
.
open paren f plus g plus h close paren times x is equal to 10 x to the 5th power minus 3 x to the 4th power plus 5 x plus cube root of x minus 6 and the domain of open paren f plus g plus h close paren times x is x is greater than or equal to 0 .

Answer 1

To find \((f + g + h)(x)\), we need to add the functions \(f(x)\), \(g(x)\), and \(h(x)\).

1. **Function Definitions**:
- \(f(x) = x^5 - 2x^4 + 5x - 11\)
- \(g(x) = 9x - 1\)
- \(h(x) = \sqrt[3]{x} + 6\)

2. **Combining the Functions**:
\[
(f + g + h)(x) = f(x) + g(x) + h(x)
\]
\[
= (x^5 - 2x^4 + 5x - 11) + (9x - 1) + (\sqrt[3]{x} + 6)
\]

Now, combining like terms:
- The \(x^5\) term: \(x^5\)
- The \(x^4\) term: \(-2x^4\)
- The \(x\) terms: \(5x + 9x = 14x\)
- The constant terms: \(-11 - 1 + 6 = -6\)

Therefore,
\[
(f + g + h)(x) = x^5 - 2x^4 + 14x - 6 + \sqrt[3]{x}
\]

3. **Domain of \((f + g + h)(x)\)**:
- The function \(f(x)\) is a polynomial, which is defined for all \(x\).
- The function \(g(x)\) is also a polynomial, defined for all \(x\).
- The function \(h(x) = \sqrt[3]{x} + 6\) is defined for all \(x\) since the cube root function is defined for all real numbers.

Since all individual functions are defined for all real numbers, the combined function \((f + g + h)(x)\) also has a domain of:
\[
(-\infty, \infty)
\]

4. **Final Results**:
\[
(f + g + h)(x) = x^5 - 2x^4 + 14x - 6 + \sqrt[3]{x}
\]
The domain of \((f + g + h)(x)\) is:
\[
(-\infty, \infty)
\]

Thus, the correct choice would be:
\((f + g + h)(x) = x^5 - 2x^4 + 14x - 6 + \sqrt[3]{x}\) and the domain of \((f + g + h)(x)\) is \((-\infty, \infty)\).