To determine which function is increasing at the highest rate, we need to analyze each option and calculate the respective rates of increase.
### Option A
The table provides the values for [tex]\( g(x) \)[/tex]:
[tex]\[
\begin{tabular}{|c|r|r|r|r|r|}
\hline
$x$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$g(x)$ & -5 & -4 & -3 & -2 & -1 \\
\hline
\end{tabular}
\][/tex]
We observe that [tex]\( g(x) \)[/tex] is increasing by 1 unit for each increase of 1 unit in [tex]\( x \)[/tex]. Thus, the rate of increase is:
[tex]\[
\text{Rate of increase for } g(x) = 1
\][/tex]
### Option B
No specific function or dataset is provided in Option B, making it impossible to determine a rate of increase from this option.
### Option C
We are given the equation [tex]\( 12x - 6y = -24 \)[/tex]. To find the rate of increase, we first rewrite this equation in the slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting with the given equation:
[tex]\[
12x - 6y = -24
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
-6y = -12x - 24
\][/tex]
[tex]\[
y = 2x + 4
\][/tex]
The slope [tex]\( m \)[/tex] of this linear equation is 2. Therefore, the rate of increase for this function is:
[tex]\[
\text{Rate of increase for } 12x - 6y = -24 \text{ is } 2
\][/tex]
### Option D
For this option, we are given that the linear function [tex]\( f \)[/tex] has an [tex]\( x \)[/tex]-intercept of 8 and a [tex]\( y \)[/tex]-intercept of -4. To find the rate of increase, we need to calculate the slope of the line passing through these intercepts.
The intercepts are points (8, 0) and (0, -4). The slope [tex]\( m \)[/tex] of the line can be calculated using these points:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-4)}{8 - 0} = \frac{4}{8} = 0.5
\][/tex]
Therefore, the rate of increase for this linear function is:
[tex]\[
\text{Rate of increase for } f(x) \text{ is } 0.5
\][/tex]
### Comparing Rates
- Option A: Rate of increase = 1
- Option B: No data given
- Option C: Rate of increase = 2
- Option D: Rate of increase = 0.5
Thus, the function with the highest rate of increase is the one given in Option C, with a rate of increase of 2.
### Conclusion
The correct answer is:
C. [tex]\( 12x - 6y = -24 \)[/tex]
