This exercise is based on the following functions.
f(x) = x2 + 4 with domain (−∞, ∞)
g(x) = x − 4 with domain (−∞, ∞)
h(x) = x + 6 with domain [20, ∞)
u(x) =

x + 20
with domain [−20, 0)
v(x) =

20 − x
with domain [0, 20]
(a)
Write a formula for the indicated function.
s = f + g
s(x) =

(b)
Give the domain. (Enter your answer using interval notation.)

(c)
Specify the function's value at the given point
a = −5,
if defined. (If an answer is undefined, enter UNDEFINED.)
s(−5) =

Let's go through each part of the problem step by step.

### Part (a): Write a formula for the function \( s(x) = f(x) + g(x) \)

You are given the following functions:
- \( f(x) = x^2 + 4 \)
- \( g(x) = x - 4 \)

Now, let's find \( s(x) \), which is the sum of \( f(x) \) and \( g(x) \):

\[
s(x) = f(x) + g(x)
\]
\[
s(x) = (x^2 + 4) + (x - 4)
\]

Simplify the expression:

\[
s(x) = x^2 + x + 4 - 4
\]
\[
s(x) = x^2 + x
\]

Thus, the formula for \( s(x) \) is:

\[
s(x) = x^2 + x
\]

### Part (b): Give the domain of \( s(x) \)

The domain of \( s(x) \) is the intersection of the domains of \( f(x) \) and \( g(x) \). Both \( f(x) \) and \( g(x) \) have the domain \( (-\infty, \infty) \), meaning they are defined for all real values of \( x \).

Therefore, the domain of \( s(x) \) is:

\[
(-\infty, \infty)
\]

### Part (c): Specify the function's value at \( a = -5 \)

Now, evaluate \( s(-5) \):

\[
s(x) = x^2 + x
\]

Substitute \( x = -5 \):

\[
s(-5) = (-5)^2 + (-5)
\]
\[
s(-5) = 25 - 5 = 20
\]

So, the value of \( s(-5) \) is:

\[
s(-5) = 20
\]

### Summary of answers:
(a) \( s(x) = x^2 + x \)

(b) The domain is \( (-\infty, \infty) \)

(c) \( s(-5) = 20 \)

Would you like further details on any step, or do you have any questions?

Here are five related questions for deeper understanding:
1. How do you find the domain when functions are combined using addition or subtraction?
2. What is the graphical interpretation of the sum of two functions?
3. Can you find the zeros of \( s(x) = x^2 + x \)?
4. How would the domain change if \( g(x) \) had a restricted domain?
5. How do function compositions differ from function addition?

**Tip**: Always remember to simplify your function before determining its domain—it helps avoid unnecessary complexity.

This exercise is based on the following functions.
f(x) = x2 + 4 with domain (−∞, ∞)
g(x) = x − 4 with domain (−∞, ∞)
h(x) = x + 6 with domain [20, ∞)
u(x) = x + 20 with domain [−20, 0)
v(x) = 20 − x with domain [0, 20]
(a) Write a formula for the indicated function.
s = f + g
s(x) =
(b) Give the domain. (Enter your answer using interval notation.)
(c) Specify the function's value at the given point
a = −5, if defined. (If an answer is undefined, enter UNDEFINED.)
s(−5) =

This problem demonstrates the addition of functions, finding the sum s(x) = x^2 + x, and determining its domain and specific values. Learn how to simplify functions and explore domains, suitable for students in Grades 9-12.

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