Question

Question
Use the functions below.

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
.
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 first need to compute \(g \cdot h\) and then subtract that from \(f(x)\).

Given the functions:

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

First, let's compute \(g(x) \cdot h(x)\):

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

Next, we need to substitute this into \(f(x)\):

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

Now, simplifying this:

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

Combining the like terms gives us:

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

Thus, we have:

\[
(f - g \cdot h)(x) = 8x + 3x|x| - 5
\]

Now, let's analyze the domain of \((f - g \cdot h)(x)\):

- The functions \(f(x)\), \(g(x)\), and \(h(x)\) have no restrictions on their domains; hence, they are defined for all real \(x\).

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

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

Putting it all together, the answer is:

\[
(f - g \cdot h)(x) = 8x + 3x|x| - 5
\]

And the domain is:

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