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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+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 (−∞,∞)
.
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+3x2+3x−|x+1|−2
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 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 x is greater than or equal to negative 1 .
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1 answer

To find \( (f - g - h)(x) \), we need to combine the functions \( f(x), g(x), \) and \( h(x) \) as follows:

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

  2. Calculate \( g(x) \): \[ g(x) = |x + 1| - 7 \]

  3. Calculate \( h(x) \): \[ h(x) = -x^4 + 2x^2 + 3 \]

Now, we will compute \( f - g - h \):

\[ (f - g - h)(x) = f(x) - g(x) - h(x) \]

Substituting the expressions we have for \( f(x) \), \( g(x) \), and \( h(x) \):

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

Simplifying this expression:

  1. Distributing the negative signs: \[ = 5x^2 + 3x - 6 - |x + 1| + 7 + x^4 - 2x^2 - 3 \]

  2. Combine like terms: \[ = x^4 + (5x^2 - 2x^2) + 3x + (7 - 6 - 3) - |x + 1| \] \[ = x^4 + 3x^2 + 3x - |x + 1| - 2 \]

So we have:

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

Next, let’s analyze the domain of \( (f - g - h)(x) \).

The functions \( f(x) \) and \( h(x) \) are polynomial functions, which are defined for all real numbers. The only constraint comes from \( g(x) = |x + 1| - 7 \), which is also defined for all real numbers. Therefore, the domain of the composed function \( (f - g - h)(x) \) is:

\[ \text{Domain of } (f - g - h)(x) = (-\infty, \infty) \]

In conclusion, the correct response is:

\[ (f - g - h)(x) = x^4 + 3x^2 + 3x - |x + 1| - 2 \quad \text{and the domain is } (-\infty, \infty). \]