Question

Use the functions below.

f(x)=x+2−−−−√3−2
g(x)=−4x+2−−−−√3+3
h(x)=−x+2−−−−√3−4
What is (f+g−h)(x)
? What is the domain of (f+g−h)(x)
?

(1 point)
Responses

(f+g−h)(x)=4x+2−−−−−√3−9
and the domain of (f+g−h)(x)
is (−∞, ∞)
.
open paren f plus g minus h close paren times x is equal to 4 cube root of x plus 2 minus 9 and the domain of open paren f plus g minus h close paren times x is open paren negative infinity comma infinity close paren .

(f+g−h)(x)=−2x+2−−−−−√3+5
and the domain of (f+g−h)(x)
is (−∞, ∞)
.
open paren f plus g minus h close paren times x is equal to negative 2 cube root of x plus 2 plus 5 and the domain of open paren f plus g minus h close paren times x is open paren negative infinity comma infinity close paren .

(f+g−h)(x)=4x+2−−−−−√3−9
and the domain of (f+g−h)(x)
is x≠−2
.
open paren f plus g minus h close paren times x is equal to 4 cube root of x plus 2 minus 9 and the domain of open paren f plus g minus h close paren times x is x is not equal to negative 2 .

(f+g−h)(x)=−2x+2−−−−−√3+5
and the domain of (f+g−h)(x)
is x≠−2
.
open paren f plus g minus h close paren times x is equal to negative 2 cube root of x plus 2 plus 5 and the domain of open paren f plus g minus h close paren times x is x is not equal to negative 2 .

Answer 1

To find \((f+g-h)(x)\), we first need to calculate \(f(x)\), \(g(x)\), and \(h(x)\) using the provided functions:

1. **Given Functions**:
- \(f(x) = \sqrt[3]{x + 2} - 2\)
- \(g(x) = \sqrt[3]{-4x + 2} + 3\)
- \(h(x) = \sqrt[3]{-x + 2} - 4\)

2. **Calculate \(f + g - h\)**:
\[
(f + g - h)(x) = f(x) + g(x) - h(x)
\]
Substituting in the values:
\[
= \left(\sqrt[3]{x + 2} - 2\right) + \left(\sqrt[3]{-4x + 2} + 3\right) - \left(\sqrt[3]{-x + 2} - 4\right)
\]
Simplifying step by step:
\[
= \sqrt[3]{x + 2} + \sqrt[3]{-4x + 2} - \sqrt[3]{-x + 2} + 5
\]

This simplifies to:
\[
= \sqrt[3]{x + 2} + \sqrt[3]{-4x + 2} - \sqrt[3]{-x + 2} + 5
\]

3. **Domain of \((f + g - h)(x)\)**:
The domain of each function is determined by finding any restrictions on the input \(x\) when doing operations such as square roots or cube roots.

In this case:
- Cube roots are defined for all real numbers, so there are no restrictions from \(f(x)\), \(g(x)\), and \(h(x)\) themselves.

Therefore, the overall domain of \((f + g - h)(x)\) is:
\[
(-\infty, \infty)
\]

Based on these calculations, \((f + g - h)(x)\) would not be any of the provided responses. However, I can confirm that:
- \((f+g-h)(x) = \sqrt[3]{x + 2} + \sqrt[3]{-4x + 2} - \sqrt[3]{-x + 2} + 5\)
- The domain of \((f + g - h)(x)\) is \((-∞, ∞)\).

If provided with a clearer choice of responses that matches this outcome or specifically indicates how \(f(x)\), \(g(x)\), and \(h(x)\) interact in terms of their compositions, I'd be able to choose accordingly.

**Please ensure you correctly check each step and calculate for confirmation, as the values and behavior of these functions should lead to finding the correct form of \((f + g - h)(x)\) in your options.**