Question

Use the functions below.

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+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 .

(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 .

Answer 1

To find \((f + g + h)(x)\), we first need to explicitly express each function:

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

Now, we combine them:

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

Combining these, we get:

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

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

- The function \(f(x) = -x^4 + 7x^2 + 6\) is a polynomial and is defined for all real numbers.
- The function \(g(x) = |x - 7| + 1\) is also defined for all real numbers.
- The function \(h(x) = \sqrt{x + 3} - 8\) requires that \(x + 3 \geq 0\), which means \(x \geq -3\).

Thus, the overall domain of \((f + g + h)(x)\) is determined by the function \(h(x)\), which gives us:

\[
\text{Domain: } x \geq -3
\]

Therefore, the final result is:

\[
(f + g + h)(x) = -x^4 + 7x^2 + |x - 7| + \sqrt{x + 3} - 1
\]
and the domain of \((f + g + h)(x)\) is \(x \geq -3\).