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Use the functions below.

f(x)=5x2+3x−6
g(x)=|x+1|−7
h(x)=−x4+2x2+3
What is (f−g−h)(x)? What is the domain of (f−g−h)(x)?

(1 point)
Responses

(f−g−h)(x)=−x4+7x2+3x+|x+1|−10 and the domain of (f−g−h)(x) is (−∞,∞).
open paren f minus g minus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus 3 x plus start absolute value x plus 1 end absolute value minus 10 and the domain of open paren f minus g minus h close paren times x is open paren negative infinity comma infinity close paren .

(f−g−h)(x)=x4+3x2+3x−|x+1|−2 and the domain of (f−g−h)(x) is (−∞,∞).
open paren f minus g minus h close paren times x is equal to x to the 4th power plus 3 x squared plus 3 x minus start absolute value x plus 1 end absolute value minus 2 and the domain of open paren f minus g minus h close paren times x is open paren negative infinity comma infinity close paren .

(f−g−h)(x)=−x4+7x2+3x+|x+1|−10 and the domain of (f−g−h)(x) is x≥−1.
open paren f minus g minus h close paren times x is equal to negative x to the 4th power plus 7 x squared plus 3 x plus start absolute value x plus 1 end absolute value minus 10 and the domain of open paren f minus g minus h close paren times x is x is greater than or equal to negative 1 .

(f−g−h)(x)=x4+3x2+3x−|x+1|−2 and the domain of (f−g−h)(x) is

1 answer

To find \((f - g - h)(x)\), we need to first express the functions \(f(x)\), \(g(x)\), and \(h(x)\) and then compute the difference.

  1. Functions:

    • \(f(x) = 5x^2 + 3x - 6\)
    • \(g(x) = |x + 1| - 7\)
    • \(h(x) = -x^4 + 2x^2 + 3\)

  2. Calculation of \((f - g - h)(x)\): \[ (f - g - h)(x) = f(x) - g(x) - h(x) \]

    Substituting the functions: \[ (f - g - h)(x) = (5x^2 + 3x - 6) - (|x + 1| - 7) - (-x^4 + 2x^2 + 3) \]

    Now, simplify this step-by-step:

    • Start with \(f(x)\): \[ 5x^2 + 3x - 6 \]

    • Subtract \(g(x)\): \[ 5x^2 + 3x - 6 - |x + 1| + 7 = 5x^2 + 3x + 1 - |x + 1| \]

    • Now subtract \(h(x)\): \[ 5x^2 + 3x + 1 - |x + 1| + x^4 - 2x^2 - 3 = x^4 + 3x^2 + 3x - |x + 1| - 2 \]

    So we find: \[ (f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2 \]

  3. Domain of \((f-g-h)(x)\): The domain of a function is typically all real numbers unless specified otherwise by pieces like absolute values which can introduce constraints. Here, for \(|x + 1|\), there are no restrictions. Thus:

    The domain of \((f - g - h)(x)\) is \((-\infty, \infty)\).

So, the final answer is: \[ (f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2 \] with the domain being \((-\infty, \infty)\).