Question

Use the functions
`f(x)` and
`g(x)` to find the domain of the indicated functions, and simplify the expressions for the domain in question.

FUNCTIONS:

`f(x) = x^4`

`g(x)=√(x+7)`

1)
`(f+g)(x)` =
`f(x)+g(x)`

2)
`(f-g)(x)` =
`f(x)-g(x)`

3)
`(fg)(x)` ->
`(f(g(x))`


POSSIBLE ANSWERS:

A) The domain of all three pairs is -∞ ≤
`x` ≤ ∞.

B) The domain for #1 and #2 is
`x` ≥ -7 and the domain for #3 is -∞ ≤
`x` ≤ ∞.

C) The domain of every pair is
`x` ≥ -7.

D) A through C are partially true.

E) None of the above OR the domain is undefined.

Answer 1

B) The domain for #1 and #2 is ≥ -7 and the domain for #3 is -∞ ≤ ≤ ∞.

Step-by-step explanation:

1.
`(f + g)(x) = f(x) + g(x)`

The domain of
`(f + g)(x)` is the intersection of the domains of
`f(x)` and
`g(x)`. Since
`f(x) = x^4` is defined for all real numbers
`\mathbb {R}`, and
`g(x) = (x + 7)^(1/2)` is defined only for
`x > = -7` (since we can't take the square root of a negative number), the domain of
`(f + g)(x)` is
`x > = -7`.

2.
`(f - g)(x) = f(x) - g(x)`

The domain of
`(f - g)(x)` is also the intersection of the domains of
`f(x)` and
`g(x)`. Since
`f(x) = x^4` is defined for all real numbers
`\mathbb {R}`, and
`g(x) = (x + 7)^(1/2)` is defined only for
`x > = -7`, the domain of
`(f - g)(x)` is also
`x > = -7`.

3.
`(fg)(x) - > (f(g(x))`

The domain of
`f(g(x))` is the set of all values of
`x` such that
`g(x)` is in the domain of
`f`. Since the domain of
`f` is all real numbers
`\mathbb {R}`, and
`g(x) = (x + 7)^(1/2)` is defined only for
`x > = -7`, the domain of
`f(g(x))` is
`x > = -7`.

Therefore, the correct answer is option:

B) The domain for 1) and 2) is
`> = -7` and the domain for 3) is -∞ ≤
`x` ≤ ∞.