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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 .

1 answer

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.