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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 will start by defining the functions \(f(x)\), \(g(x)\), and \(h(x)\):

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

The expression for \((f - g - h)(x)\) is given by: \[ (f - g - h)(x) = f(x) - g(x) - h(x) \]

Now, we can substitute the functions in:

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

  2. Subtract \(g(x)\): \[ g(x) = |x + 1| - 7 \] So, \[ f(x) - g(x) = (5x^2 + 3x - 6) - (|x + 1| - 7) = 5x^2 + 3x - 6 - |x + 1| + 7 \] Which simplifies to: \[ f(x) - g(x) = 5x^2 + 3x + 1 - |x + 1| \]

  3. Now subtract \(h(x)\): \[ h(x) = -x^4 + 2x^2 + 3 \] So, \[ (f - g)(x) - h(x) = (5x^2 + 3x + 1 - |x + 1|) - (-x^4 + 2x^2 + 3) \] That becomes: \[ f(x) - g(x) - h(x) = 5x^2 + 3x + 1 - |x + 1| + x^4 - 2x^2 - 3 \] Simplifying further: \[ = x^4 + (5x^2 - 2x^2) + 3x + (1 - 3) - |x + 1| \] \[ = x^4 + 3x^2 + 3x - |x + 1| - 2 \]

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

Now let’s find the domain of \((f - g - h)(x)\):

  • Since \(f(x)\), \(g(x)\), and \(h(x)\) are all polynomials or piecewise continuous functions, their domains are all real numbers.
  • The absolute value function \(g(x) = |x + 1| - 7\) does not impose any restrictions as it is defined for all \(x\).

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

\[ (-\infty, \infty) \]

Final Answer:

\((f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2\) and the domain of \((f - g - h)(x)\) is \((-∞, ∞)\).