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)=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 .

(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)=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)=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 .

Answer 1

To find \((f + g + h)(x)\), we will add the three functions \(f(x)\), \(g(x)\), and \(h(x)\) together.

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

Let's add the three functions:

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

\[
= (x^5 - 2x^4 + 5x - 11) + (9x - 1) + (\sqrt[3]{x} + 6)
\]

Now combine like terms:

\[
= x^5 - 2x^4 + 5x + 9x - 11 - 1 + 6 + \sqrt[3]{x}
\]

Combine the \(x\) terms and constant terms:

\[
= x^5 - 2x^4 + (5x + 9x) + \sqrt[3]{x} + (-11 - 1 + 6)
\]

\[
= x^5 - 2x^4 + 14x + \sqrt[3]{x} - 6
\]

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

Next, we need to determine the domain of \((f + g + h)(x)\):

- The function \(f(x)\) is a polynomial, 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 can take any real number.

Since all three functions are defined for all real numbers, the domain of \((f + g + h)(x)\) is:

\[
(-\infty, \infty)
\]

Therefore, the answer is:
\[
(f + g + h)(x) = x^5 - 2x^4 + 14x + \sqrt[3]{x} - 6 \quad \text{and the domain is} \quad (-\infty, \infty).
\]