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+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 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 (−∞, ∞)
.
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+3−−−−−√−1
and the domain of (f+g+h)(x)
is (−∞, ∞)
.

Answer 1

To find \((f+g+h)(x)\), we will first write out each function and then combine them.

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

2. Now we can combine these to find \((f + g + h)(x)\):

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

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

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

- The function \(f(x)\) has no restrictions.
- The function \(g(x) = |x - 7| + 1\) also has no restrictions on its domain since absolute value functions are defined for all real numbers.
- The function \(h(x) = \sqrt{x + 3} - 8\) requires that the expression inside the square root be non-negative:
\[
x + 3 \geq 0 \implies x \geq -3
\]

Thus, the overall domain of \((f + g + h)(x)\) is:

\[
x \geq -3
\]

In interval notation, the domain is:
\[
[-3, \infty)
\]

So the final answers are:
- \((f + g + h)(x) = -x^4 + 7x^2 + |x - 7| + \sqrt{x + 3} - 1\)
- The domain of \((f + g + h)(x)\) is \(x \geq -3\).