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Use the functions below.

f(x)=x−1−−−−√3+7
g(x)=|x+4|
What is (fg)(x)
? What is the domain of (fg)(x)
?

(1 point)
Responses

(fg)(x)=x−1√3+7|x+4|
and the domain of (fg)(x)
is (−∞,∞)
.
open paren f over g close paren of x is equal to the fraction with numerator cube root of x minus 1 plus 7 and denominator start absolute value x plus 4 end absolute value and the domain of open paren f over g close paren of x is open paren negative infinity comma infinity close paren .

(fg)(x)=x−1√3+7|x+4|
and the domain of (fg)(x)
is x≠−4
.
open paren f over g close paren of x is equal to the fraction with numerator cube root of x minus 1 plus 7 and denominator start absolute value x plus 4 end absolute value and the domain of open paren f over g close paren of x is x is not equal to negative 4 .

(fg)(x)=|x+4|x−1√3+7
and the domain of (fg)(x)
is (−∞,∞)
.
open paren f over g close paren of x is equal to the fraction with numerator start absolute value x plus 4 end absolute value and denominator cube root of x minus 1 plus 7 and the domain of open paren f over g close paren of x is open paren negative infinity comma infinity close paren .

(fg)(x)=|x+4|x−1√3+7
and the domain of (fg)(x)
is x≠−4
.
open paren f over g close paren of x is equal to the fraction with numerator start absolute value x plus 4 end absolute value and denominator cube root of x minus 1 plus 7 and the domain of open paren f over g close paren of x is x is not equal to negative 4 .

1 answer

To clarify the notation and interpretation of function composition for \( (fg)(x) \):

  1. Function Definitions:

    • \( f(x) = \sqrt[3]{x - 1} + 7 \)
    • \( g(x) = |x + 4| \)

  2. Composition of Functions: When we say \( (fg)(x) \), we usually mean \( f(g(x)) \). In this case, it would mean substituting \( g(x) \) into \( f(x) \).

  3. Calculating \( f(g(x)) \): \[ f(g(x)) = f(|x + 4|) = \sqrt[3]{|x + 4| - 1} + 7 \]

  4. Determining the Domain:

    • \( g(x) \) is defined for all \( x \), thus the domain of \( g(x) \) is \( (-\infty, \infty) \).
    • \( f(x) \) is defined for all real numbers since the cube root function and constant addition are defined everywhere.
    • Therefore, \( f(g(x)) \) is also defined for all real numbers.

Thus, we have: \[ (fg)(x) = \sqrt[3]{|x + 4| - 1} + 7 \] And the domain of \( (fg)(x) \) is \( (-\infty, \infty) \).

Among the provided responses, the correct one regarding the composition function \( (fg)(x) \) and its domain is:

\( (fg)(x) = \sqrt[3]{|x + 4| - 1} + 7 \) and the domain of \( (fg)(x) \) is \( (-\infty, \infty) \).

Since this exact option isn't listed, none of the provided options satisfy the reasoning accurately. If you are looking for a pick among the given responses, check for the closest match, but the representation as \( (fg)(x) \) would primarily state as stated above.