Use the functions below.

Question

f(x)=2x−5
g(x)=|x|+2
h(x)=−3x
What is (f−g⋅h)(x)
? What is the domain of (f−g⋅h)(x)
?

(1 point)
Responses

(f−g⋅h)(x)=2x+3x|x|+6x−5
and the domain of (f−g⋅h)(x)
is (−∞, ∞)
.
open paren f minus g times h close paren times x is equal to 2 to the x th power plus 3 x divides x divides plus 6 x minus 5 and the domain of open paren f minus g times h close paren times x is open paren negative infinity comma infinity close paren .

(f−g⋅h)(x)=−3⋅2xx+3x|x|+21x
and the domain of (f−g⋅h)(x)
is x≠0
.
open paren f minus g times h close paren times x is equal to negative 3 times 2 to the x th power x plus 3 x divides x divides plus 21 x and the domain of open paren f minus g times h close paren times x is x is not equal to 0 .

(f−g⋅h)(x)=−3⋅2xx+3x|x|+21x
and the domain of (f−g⋅h)(x)
is (−∞, ∞)
.
open paren f minus g times h close paren times x is equal to negative 3 times 2 to the x th power x plus 3 x divides x divides plus 21 x and the domain of open paren f minus g times h close paren times x is open paren negative infinity comma infinity close paren .

(f−g⋅h)(x)=2x+3x|x|+6x−5
and the domain of (f−g⋅h)(x)
is x≠0
.

Answer 1

To determine \((f - g \cdot h)(x)\) and its domain, we need to calculate \(g \cdot h\) first, and then subtract it from \(f\).

1. Calculate \(g(x) \cdot h(x)\):

   \(g(x) = |x| + 2\)

   \(h(x) = -3x\)

   So: \[ g(x) \cdot h(x) = (|x| + 2)(-3x) = -3x|x| - 6x \]

2. Now calculate \(f(x) - g(x) \cdot h(x)\):

   \(f(x) = 2x - 5\)

   Thus: \[ (f - g \cdot h)(x) = f(x) - g(x) \cdot h(x) = (2x - 5) - (-3x|x| - 6x) \]

   Simplifying this gives: \[ (f - g \cdot h)(x) = 2x - 5 + 3x|x| + 6x \] \[ (f - g \cdot h)(x) = 2x + 6x + 3x|x| - 5 \] \[ (f - g \cdot h)(x) = 8x + 3x|x| - 5 \]

3. Determine the domain:

The functions \(f(x)\), \(g(x)\), and \(h(x)\) are all defined for all real numbers. Therefore, the domain of \((f - g \cdot h)(x)\) is: \[ (-\infty, \infty) \]

Thus, the final answer is:

\[ (f - g \cdot h)(x) = 8x + 3x|x| - 5 \quad \text{and the domain is } (-\infty, \infty) \]