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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 .

1 answer

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 \((-\infty, \infty)\).