Question

1. terry wants to rent a karaoke machine for a family reunion. if she rents it from mikes music, there is a $75.00 deposit and $12.50 cost per day. independent (x) is: dependent (y) is: function: f(x) = continuous or discrete why? domain: range: function table: as x → ∞, y → as x → -∞, y →

Explanation:

Step1: Identify variables

Independent ($x$) = number of rental days; Dependent ($y$) = total rental cost

Step2: Define the function

Fixed deposit + daily cost: $f(x) = 12.50x + 75$

Step3: Classify function type

Days are whole positive numbers, so discrete.

Step4: Explain discrete classification

You can't rent a fraction of a day.

Step5: Determine domain

$x$ = non-negative whole numbers: $\{0, 1, 2, 3, ...\}$

Step6: Determine range

Calculate costs for domain values: $\{75, 87.50, 100, 112.50, ...\}$

Step7: Complete function table

Calculate $f(x)$ for $x=0,1,2,3$:

• $x=0$: $f(0)=12.50(0)+75=75$
• $x=1$: $f(1)=12.50(1)+75=87.50$
• $x=2$: $f(2)=12.50(2)+75=100$
• $x=3$: $f(3)=12.50(3)+75=112.50$

Step8: Analyze end behavior

As $x\to\infty$, $12.50x$ grows infinitely: $y\to\infty$
As $x\to-\infty$, negative days are invalid, so no real meaning, but mathematically $y\to-\infty$ (not practical here)

Answer:

Independent ($x$) is: number of rental days
Dependent ($y$) is: total rental cost
Function: $f(x) = 12.50x + 75$
Continuous or Discrete: Discrete
Why? Days are whole positive numbers
Domain: $\{0, 1, 2, 3, ...\}$
Range: $\{75, 87.50, 100, 112.50, ...\}$
Function Table:

$x$ (day)	$y$ (cost)
1	87.50
2	100.00
3	112.50

As $x \to \infty$, $y \to \infty$
As $x \to -\infty$, $y \to -\infty$ (not practical for this context)