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)=−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)=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 (−∞, ∞)
.
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
.
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 x is not equal to 0 .

Answer 1

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

Given functions:

• \( f(x) = 2x - 5 \)
• \( g(x) = |x| + 2 \)
• \( h(x) = -3x \)

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

Now simplify \( g \cdot h \): \[ g(x) \cdot h(x) = -3x|x| - 6x \]

Next, we subtract this from \( f(x) \): \[ (f - g \cdot h)(x) = f(x) - (g(x) \cdot h(x)) = (2x - 5) - (-3x|x| - 6x) = 2x - 5 + 3x|x| + 6x = (2x + 6x) + 3x|x| - 5 = 8x + 3x|x| - 5 \]

So the final expression is: \[ (f - g \cdot h)(x) = 3x|x| + 8x - 5 \]

Next, let's determine the domain of \( (f - g \cdot h)(x) \).

• The functions \( f(x) \), \( g(x) \), and \( h(x) \) themselves are defined for all real numbers, so there are no restrictions from these functions individually.

Therefore, the domain of \( (f - g \cdot h)(x) \) is all real numbers: \[ \text{Domain} = (-\infty, \infty) \]

To summarize:

• \( (f - g \cdot h)(x) = 3x|x| + 8x - 5 \)
• Domain of \( (f - g \cdot h)(x) = (-\infty, \infty) \)