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 (−∞, ∞)
.
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 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 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 (−∞, ∞)
.

Answer 1

To find \((f+g+h)(x)\), we need to add the three functions together:

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

Let's add them step by step.

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

Substituting the functions in:

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

Now, combine like terms:

1. The \(x^5\) term: \(x^5\)
2. The \(x^4\) term: \(-2x^4\)
3. The \(x\) terms: \(5x + 9x = 14x\)
4. The constant terms: \(-11 - 1 + 6 = -6\)
5. The cube root term: \(\sqrt[3]{x}\)

So, we get:

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

Next, we determine the domain of \((f + g + h)(x)\). The function \(g(x) = 9x - 1\) and \(f(x) = x^5 - 2x^4 + 5x - 11\) are polynomials and are defined for all real \(x\). The term \(h(x) = \sqrt[3]{x} + 6\) is also defined for all \(x\) since the cube root function is defined for all real numbers.

Therefore, the combined domain of \( (f+g+h)(x) \) is:

\[
\text{Domain: } (-\infty, \infty)
\]

Putting it all together, the answer is:

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