Question

Determine the domain of the function.

f as a function of x is equal to the square root of x plus four divided by x plus two times x minus five.

All real numbers except -4, -2, and 5
x ≥ -4, x ≠ -2, x ≠ 5
All real numbers
x ≥ 0

Answer 1

The domain of the function is:

x ≥ -4, x ≠ -2, x ≠ 5

Explanation:

Domain of a function--

A domain of a function is the set of all the input values i.e. the x-values for which a function is defined.

Here we are given a function f(x) as follows:

`f(x)=(√(x+4))/((x+2)(x-5))`

Now, we know that the domain of the function depends on the denominator as well as domain of the square root function.

Since, a rational function is defined at the points excluding the zero of the denominator function.

i.e.

`x\\eq -2,5`

Also, a square root function is defined for all the non-negative values of the function inside the radical sign.

i.e.

`x+4\geq 0\\\\i.e.\\\\x\geq -4`

Hence, the domain is:

x ≥ -4, x ≠ -2, x ≠ 5

Answer 2

The marked answer is correct.

Explanation:

The domain of the function is limited to those values of x for which the function is defined. The value under the radical must be non-negative (x ≥ -4), and the denominator must not be zero (x ≠ -2 or 5).

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For the purpose here, the square root of a negative number is undefined, as is division by zero.