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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)=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 .

(f−g⋅h)(x)=−3⋅2xx+3x|x|+21x
and the domain of (f−g⋅h)(x)
is (−∞, ∞)
.

1 answer

To find \((f - g \cdot h)(x)\), we need to evaluate the expression as follows:

  1. Start with the functions:

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

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

  3. Now substitute \(g \cdot h\) back into \(f - g \cdot h\): \[ (f - g \cdot h)(x) = f(x) - g \cdot h(x) = (2x - 5) - (-3x \cdot |x| - 6x) \] \[ = 2x - 5 + 3x \cdot |x| + 6x \] \[ = (2x + 6x + 3x \cdot |x| - 5) = (8x + 3x \cdot |x| - 5) \]

Thus, we get: \[ (f - g \cdot h)(x) = 8x + 3x \cdot |x| - 5 \]

  1. Now for the domain of \((f - g \cdot h)(x)\):
    • Both \(f(x)\) and \(h(x)\) are defined for all real numbers.
    • \(g(x)\) is also defined for all real numbers since absolute value is defined everywhere.
    • The product \(g \cdot h\) is also defined for all \(x\).

Therefore, the domain of \((f - g \cdot h)(x)\) is \((-\infty, \infty)\).

In summary:

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

And the domain of \((f - g \cdot h)(x)\) is \((-\infty, \infty)\).