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+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 (−∞, ∞)
.

Answer 1

To find \((f + g + h)(x)\) and its domain, we will start by combining the functions \(f(x)\), \(g(x)\), and \(h(x)\).

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

First, let's find \((f + g + h)(x)\):

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

Substituting in the expressions for each function:

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

Now, we simplify this expression:

Combining all the constant terms:
\[
6 + 1 - 8 = -1
\]

So, we can rewrite it as:
\[
(f + g + h)(x) = -x^4 + 7x^2 + |x - 7| + \sqrt{x + 3} - 1
\]

Next, we determine the domain of \((f + g + h)(x)\).

1. **Domain of \(f(x)\)**: This function is a polynomial, so its domain is all real numbers: \((-∞, ∞)\).

2. **Domain of \(g(x)\)**: This function involves an absolute value, which is defined for all real numbers, so its domain is also: \((-∞, ∞)\).

3. **Domain of \(h(x)\)**: This function involves a square root. The expression under the square root must be non-negative:
\[
x + 3 \geq 0 \implies x \geq -3
\]
Thus, the domain of \(h(x)\) is: \([-3, ∞)\).

To find the overall domain of \((f + g + h)(x)\), we take the intersection of the domains of \(f(x)\), \(g(x)\), and \(h(x)\):
- The domain of \(f\) and \(g\) is all real numbers, and the domain of \(h\) is \([-3, ∞)\). Therefore, the domain of \((f + g + h)(x)\) becomes:
\[
[-3, ∞)
\]

So 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 \(x \geq -3\), or in interval notation: \([-3, ∞)\).