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The final cost of a sale item is determined by multiplying the price on the tag by [tex]$75 \%$[/tex]. Which best describes the function that represents this situation?

Item Cost
\begin{tabular}{|c|c|}
\hline Price on the Tag, [tex]$x$[/tex] & Final Cost \\
\hline \[tex]$10 & \$[/tex]0.75(10) \\
\hline \[tex]$20 & \$[/tex]0.75(20) \\
\hline \[tex]$30 & \$[/tex]0.75(30) \\
\hline \[tex]$40 & \$[/tex]0.75(40) \\
\hline
\end{tabular}

A. It is linear because the ratio of the change in the final cost compared to the rate of change in the price tag is constant.
B. It is linear because the function is continuous.
C. It is nonlinear because the final cost is determined by multiplying each price tag by 0.75.
D. It is nonlinear because the price tag and final cost columns do not have the same common difference.

Sagot :

GinnyAnsweridn
To understand which description best represents the function, we first examine the relationship between the price on the tag and the final cost.

Given data: The final cost is determined by multiplying the price on the tag by 75%.

Let's analyze the table provided:

[tex]\[ \begin{array}{|c|c|} \hline \text{Price on the Tag, } x & \text{Final Cost} \\ \hline \$10 & 0.75 \times 10 = \$7.50 \\ \hline \$20 & 0.75 \times 20 = \$15.00 \\ \hline \$30 & 0.75 \times 30 = \$22.50 \\ \hline \$40 & 0.75 \times 40 = \$30.00 \\ \hline \end{array} \][/tex]

We can see a linear relationship here because the final cost is directly proportional to the price on the tag. This can be represented by the equation [tex]\( y = 0.75x \)[/tex], where [tex]\( y \)[/tex] is the final cost and [tex]\( x \)[/tex] is the price on the tag.

To show that this function is linear, let's consider the following aspects:

1. Proportionality:
- For each increment of [tex]\(\$10\)[/tex] in the price on the tag, the increment in the final cost is constant at [tex]\(\$7.50\)[/tex].
- Increment from [tex]\(\$10\)[/tex] to [tex]\(\$20: \Delta x = \$10\)[/tex], [tex]\(\Delta y = \$7.50\)[/tex]
- Increment from [tex]\(\$20\)[/tex] to [tex]\(\$30: \Delta x = \$10\)[/tex], [tex]\(\Delta y = \$7.50\)[/tex]
- Increment from [tex]\(\$30\)[/tex] to [tex]\(\$40: \Delta x = \$10\)[/tex], [tex]\(\Delta y = \$7.50\)[/tex]
- The ratio of change [tex]\(\frac{\Delta y}{\Delta x} = \frac{7.50}{10} = 0.75\)[/tex] remains constant, indicating a linear relationship.

2. Linearity:
- A function is linear if it can be described by the equation [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- In this case, [tex]\( y = 0.75x + 0 \)[/tex], confirming it is a linear function with a slope of 0.75 and an intercept of 0.

3. Continuity:
- The function [tex]\( y = 0.75x \)[/tex] is continuous, and there are no breaks or gaps in the graph.

Based on this analysis, the best description is:

It is linear because the ratio of the change in the final cost compared to the rate of change in the price tag is constant.
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