Question

question 1
to manufacture boxes, it costs $970 plus $2 for each box produced. the boxes are then sold for $4 each.
a) find an expression for the production cost of x boxes.
b) find an expression for the revenue earned by selling x boxes.
c) what is the breakeven point?
d) for what values of x will the revenue be greater than the production cost?

question 2
what function is both even and odd?

Explanation:

Formulate linear cost and revenue models

Using the Linear Equations knowledge point, we construct the cost and revenue functions for \(x\) boxes.
The fixed cost is \$970 and the variable cost is \$2 per box:

$$C(x) = 2x + 970$$

The selling price is \$4 per box:

$$R(x) = 4x$$

Determine the breakeven point

The breakeven point occurs when total revenue equals total production cost:

$$R(x) = C(x)$$

$$4x = 2x + 970$$

$$2x = 970$$

$$x = 485$$

Solve the revenue inequality

To find when revenue is greater than production cost:

$$R(x) > C(x)$$

$$4x > 2x + 970$$

$$2x > 970$$

$$x > 485$$

Analyze even and odd function definitions

An even function satisfies \(f(-x) = f(x)\) and an odd function satisfies \(f(-x) = -f(x)\).
If a function is both even and odd, then for all \(x\) in its domain:

$$f(x) = -f(x)$$

$$2f(x) = 0$$

$$f(x) = 0$$

Answer:

Question 1

a) \(2x + 970\)
b) \(4x\)
c) 485 boxes
d) \(x > 485\)

Question 2

The zero function, \(f(x) = 0\).