Question

Consider the following functions,S(x) = -15 - x and g(x) = 4 - *Step 2 of 2: Find the domain of (8 - D«). Express your answer in interval notationAnswerDomain in interval notation:

Answer 1

Given the functions:

`\begin{gathered} f(x)=-√(5-x) \\ \\ and \\ \\ g(x)=4-x \end{gathered}`

Let's find the domain of (g - f)(x) in interval notation.

To solve for (g - f)(x), let's solve for g(x) - f(x).

Subtract f(x) from g(x).

We have:

`\begin{gathered} (g-f)(x)=g(x)-f(x)=(4-x)-(-√(5-x)) \\ \\ (g-f)(x)=(4-x)-(-√(5-x)) \end{gathered}`

Solving further:

Apply distributive property and remove the parentheses.

`(g-f)(x)=4-x+√(5-x)`

Now, let's find the domain.

The domain is the set of possible values of x which makes the function defined.

To find the domain set the values in the radicand greater or equal to zero and solve for x.

`\begin{gathered} 5-x\ge0 \\ \\ \text{ Subtract 5 from both sides:} \\ -5+5-x\ge0-5 \\ \\ -x\ge-5 \\ \end{gathered}`

Divide both sides by -1:

`\begin{gathered} (-x)/(-1)\ge(-5)/(-1) \\ \\ x\leq5 \end{gathered}`

Therefore, the domain is:

x ≤ 5

In interval notation, the domain is:

`(-\infty,5]`

ANSWER:

`(-\infty,5]`