Use the functions below.

Question

f(x)=−x4+7x2+6
g(x)=|x−7|+1
h(x)=x+3−−−−√−8
What is (f+g+h)(x)
? What is the domain of (f+g+h)(x)
?

(1 point)
Responses

(f+g+h)(x)=−x4+7x2+|x−7|+x+3−−−−−√−1
and the domain of (f+g+h)(x)
is x≥−3
.
open paren f plus g plus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus start absolute value x minus 7 end absolute value plus square root of x plus 3 end root minus 1 and the domain of open paren f plus g plus h close paren times x is x is greater than or equal to negative 3 .

(f+g+h)(x)=−x4+7x2+|x−7|+x+2−−−−−√
and the domain of (f+g+h)(x)
is x≥−3
.
open paren f plus g plus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus start absolute value x minus 7 end absolute value plus square root of x plus 2 end root and the domain of open paren f plus g plus h close paren times x is x is greater than or equal to negative 3 .

(f+g+h)(x)=−x4+7x2+|x−7|+x+3−−−−−√−1
and the domain of (f+g+h)(x)
is (−∞, ∞)
.
open paren f plus g plus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus start absolute value x minus 7 end absolute value plus square root of x plus 3 end root minus 1 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)=−x4+7x2+|x−7|+x+2−−−−−√
and the domain of (f+g+h)(x)
is (−∞, ∞)
.
open paren f plus g plus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus start absolute value x minus 7 end absolute value plus square root of x plus 2 end root 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 determine \((f + g + h)(x)\), we need to combine the functions \(f(x)\), \(g(x)\), and \(h(x)\):

1. Functions given:

   • \(f(x) = -x^4 + 7x^2 + 6\)
   • \(g(x) = |x - 7| + 1\)
   • \(h(x) = \sqrt{x + 3} - 8\)

2. Combine the functions: \[ (f + g + h)(x) = f(x) + g(x) + h(x) \] \[ = (-x^4 + 7x^2 + 6) + (|x - 7| + 1) + (\sqrt{x + 3} - 8) \] \[ = -x^4 + 7x^2 + 6 + |x - 7| + 1 + \sqrt{x + 3} - 8 \] \[ = -x^4 + 7x^2 + |x - 7| + \sqrt{x + 3} - 1 \]

Thus, \[ (f + g + h)(x) = -x^4 + 7x^2 + |x - 7| + \sqrt{x + 3} - 1 \]

3. Determine the domain:
   • The domain of \(f(x)\) (a polynomial) is all real numbers, \((-∞, ∞)\).
   • The domain of \(g(x) = |x - 7| + 1\) is also all real numbers, \((-∞, ∞)\).
   • The function \(h(x) = \sqrt{x + 3} - 8\) requires that the argument of the square root is non-negative: \[ x + 3 \geq 0 \quad \Rightarrow \quad x \geq -3 \] Therefore, the domain of \(h(x)\) is \([-3, ∞)\).

Since \(h(x)\) restricts the overall domain, the domain of \((f + g + h)(x)\) is: \[ [-3, ∞) \]

Final Answer

\[ (f + g + h)(x) = -x^4 + 7x^2 + |x - 7| + \sqrt{x + 3} - 1 \] The domain of \((f + g + h)(x)\) is \(x \geq -3\) (or in interval notation, \([-3, ∞)\)).