Question

Function Operations Unit Test
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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)=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)=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
.

Answer 1

To find \( (f + g + h)(x) \) for the given functions:

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

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

Combine like terms:
\[
= x^5 - 2x^4 + (5x + 9x) + (-11 - 1 + 6) + \sqrt[3]{x}
\]
\[
= x^5 - 2x^4 + 14x - 6 + \sqrt[3]{x}
\]

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

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

Therefore, the combination \( (f + g + h)(x) \) is also defined for all \( x \in \mathbb{R} \).

Thus, the final answer is:

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

The correct response is:
**(f + g + h)(x) = x^5 - 2x^4 + 14x + \sqrt[3]{x} - 6 and the domain of (f + g + h)(x) is (-∞, ∞).**