This exercise is based on the following functions.
f(x) = x2 + 4 with domain (−∞, ∞)
g(x) = x − 4 with domain (−∞, ∞)
h(x) = x + 7 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 = −6,
if defined. (If an answer is undefined, enter UNDEFINED.)
s(−6) =

Let's go step by step through the problem.

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

We are given:
- \( f(x) = x^2 + 4 \)
- \( g(x) = x - 4 \)

To find \( s(x) \), we simply add \( f(x) \) and \( g(x) \):

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

Now, simplify the equation:

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

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

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

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

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

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

Since there are no restrictions (like division by zero or square roots of negative numbers), the domain of \( s(x) \) is also all real numbers:

\[
\text{Domain of } s(x) = (-\infty, \infty)
\]

### Part (c): Find \( s(-6) \)

Now we substitute \( x = -6 \) into the formula for \( s(x) \):

\[
s(-6) = (-6)^2 + (-6)
\]

\[
s(-6) = 36 - 6
\]

\[
s(-6) = 30
\]

So, \( s(-6) = 30 \).

### Summary of the answers:
- (a) The formula for \( s(x) \) is \( s(x) = x^2 + x \).
- (b) The domain of \( s(x) \) is \( (-\infty, \infty) \).
- (c) The value \( s(-6) = 30 \).

Would you like further details or clarification on any of these parts?

Here are 5 related questions to explore:
1. How would the domain change if \( f(x) \) or \( g(x) \) included square roots?
2. What would happen to the domain if \( g(x) \) had a denominator?
3. Can you find \( s(0) \) for the same function?
4. How would you differentiate \( s(x) \)?
5. What is the range of \( s(x) \)?

**Tip:** Always check the domain of individual functions before adding them!

This exercise is based on the following functions.
f(x) = x^2 + 4 with domain (−∞, ∞)
g(x) = x − 4 with domain (−∞, ∞)
h(x) = x + 7 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 = −6, if defined. (If an answer is undefined, enter UNDEFINED.)
s(−6) =

Learn how to add two functions, f(x) = x^2 + 4 and g(x) = x - 4, to find s(x) = x^2 + x. Understand how to determine the domain and calculate s(-6). Suitable for students in Grades 9-12.

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