Answer :
Let's analyze the given information:
1. We have an exponential function [tex]\( f(x) \)[/tex] represented by:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 \\ \hline f(x) & -62 & -30 & -14 & -6 \\ \hline \end{tabular} \][/tex]
2. We have another function [tex]\( g(x) \)[/tex] given by:
[tex]\[ g(x) = -20 \left( \frac{1}{2} \right)^x + 10 \][/tex]
We are to compare these two functions on the interval [tex]\([-2, 1]\)[/tex] and choose the correct statement among the given options.
### Step-by-step Analysis:
#### Step 1: Check if the functions are increasing
For [tex]\( f(x) \)[/tex]:
- [tex]\( f(-2) = -62 \)[/tex]
- [tex]\( f(-1) = -30 \)[/tex]
- [tex]\( f(0) = -14 \)[/tex]
- [tex]\( f(1) = -6 \)[/tex]
We compare each subsequent value:
- [tex]\( -30 > -62 \)[/tex]
- [tex]\( -14 > -30 \)[/tex]
- [tex]\( -6 > -14 \)[/tex]
Since each value of [tex]\( f(x) \)[/tex] is greater than the previous value, [tex]\( f(x) \)[/tex] is increasing.
For [tex]\( g(x) \)[/tex]:
We need to evaluate [tex]\( g(x) \)[/tex] at the same points:
[tex]\[ \begin{align*} g(-2) &= -20 \left( \frac{1}{2} \right)^{-2} + 10 = -20 \cdot 4 + 10 = -80 + 10 = -70 \\ g(-1) &= -20 \left( \frac{1}{2} \right)^{-1} + 10 = -20 \cdot 2 + 10 = -40 + 10 = -30 \\ g(0) &= -20 \left( \frac{1}{2} \right)^{0} + 10 = -20 \cdot 1 + 10 = -20 + 10 = -10 \\ g(1) &= -20 \left( \frac{1}{2} \right)^{1} + 10 = -20 \cdot \frac{1}{2} + 10 = -10 + 10 = 0 \end{align*} \][/tex]
We compare each subsequent value:
- [tex]\( -30 > -70 \)[/tex]
- [tex]\( -10 > -30 \)[/tex]
- [tex]\( 0 > -10 \)[/tex]
Since each value of [tex]\( g(x) \)[/tex] is greater than the previous value, [tex]\( g(x) \)[/tex] is also increasing.
#### Step 2: Check if the functions are negative
- [tex]\( f(x) \)[/tex] values are [tex]\(-62, -30, -14, -6\)[/tex] which are all negative.
- [tex]\( g(x) \)[/tex] values are [tex]\(-70, -30, -10, 0\)[/tex], and at [tex]\( x=1 \)[/tex], [tex]\( g(x) = 0 \)[/tex], which is not negative.
#### Step 3: Compare rates of increase
Average rate of increase for [tex]\( f \)[/tex] over [tex]\([-2, 1]\)[/tex]:
[tex]\[ \frac{f(1) - f(-2)}{1 - (-2)} = \frac{-6 - (-62)}{3} = \frac{56}{3} \approx 18.67 \][/tex]
Average rate of increase for [tex]\( g \)[/tex] over [tex]\([-2, 1]\)[/tex]:
[tex]\[ \frac{g(1) - g(-2)}{1 - (-2)} = \frac{0 - (-70)}{3} = \frac{70}{3} \approx 23.33 \][/tex]
Since [tex]\( 23.33 \)[/tex] (rate of increase of [tex]\( g \)[/tex]) is greater than [tex]\( 18.67 \)[/tex] (rate of increase of [tex]\( f \)[/tex]), [tex]\( g \)[/tex] has a faster average rate of increase than [tex]\( f \)[/tex].
### Conclusion
Based on the provided analysis:
- Both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing.
- [tex]\( g \)[/tex] increases at a faster average rate than [tex]\( f \)[/tex].
Thus, the correct statement is:
[tex]\[ \boxed{\text{C. Both functions are increasing, but function } g \text{ increases at a faster average rate.}} \][/tex]
1. We have an exponential function [tex]\( f(x) \)[/tex] represented by:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 \\ \hline f(x) & -62 & -30 & -14 & -6 \\ \hline \end{tabular} \][/tex]
2. We have another function [tex]\( g(x) \)[/tex] given by:
[tex]\[ g(x) = -20 \left( \frac{1}{2} \right)^x + 10 \][/tex]
We are to compare these two functions on the interval [tex]\([-2, 1]\)[/tex] and choose the correct statement among the given options.
### Step-by-step Analysis:
#### Step 1: Check if the functions are increasing
For [tex]\( f(x) \)[/tex]:
- [tex]\( f(-2) = -62 \)[/tex]
- [tex]\( f(-1) = -30 \)[/tex]
- [tex]\( f(0) = -14 \)[/tex]
- [tex]\( f(1) = -6 \)[/tex]
We compare each subsequent value:
- [tex]\( -30 > -62 \)[/tex]
- [tex]\( -14 > -30 \)[/tex]
- [tex]\( -6 > -14 \)[/tex]
Since each value of [tex]\( f(x) \)[/tex] is greater than the previous value, [tex]\( f(x) \)[/tex] is increasing.
For [tex]\( g(x) \)[/tex]:
We need to evaluate [tex]\( g(x) \)[/tex] at the same points:
[tex]\[ \begin{align*} g(-2) &= -20 \left( \frac{1}{2} \right)^{-2} + 10 = -20 \cdot 4 + 10 = -80 + 10 = -70 \\ g(-1) &= -20 \left( \frac{1}{2} \right)^{-1} + 10 = -20 \cdot 2 + 10 = -40 + 10 = -30 \\ g(0) &= -20 \left( \frac{1}{2} \right)^{0} + 10 = -20 \cdot 1 + 10 = -20 + 10 = -10 \\ g(1) &= -20 \left( \frac{1}{2} \right)^{1} + 10 = -20 \cdot \frac{1}{2} + 10 = -10 + 10 = 0 \end{align*} \][/tex]
We compare each subsequent value:
- [tex]\( -30 > -70 \)[/tex]
- [tex]\( -10 > -30 \)[/tex]
- [tex]\( 0 > -10 \)[/tex]
Since each value of [tex]\( g(x) \)[/tex] is greater than the previous value, [tex]\( g(x) \)[/tex] is also increasing.
#### Step 2: Check if the functions are negative
- [tex]\( f(x) \)[/tex] values are [tex]\(-62, -30, -14, -6\)[/tex] which are all negative.
- [tex]\( g(x) \)[/tex] values are [tex]\(-70, -30, -10, 0\)[/tex], and at [tex]\( x=1 \)[/tex], [tex]\( g(x) = 0 \)[/tex], which is not negative.
#### Step 3: Compare rates of increase
Average rate of increase for [tex]\( f \)[/tex] over [tex]\([-2, 1]\)[/tex]:
[tex]\[ \frac{f(1) - f(-2)}{1 - (-2)} = \frac{-6 - (-62)}{3} = \frac{56}{3} \approx 18.67 \][/tex]
Average rate of increase for [tex]\( g \)[/tex] over [tex]\([-2, 1]\)[/tex]:
[tex]\[ \frac{g(1) - g(-2)}{1 - (-2)} = \frac{0 - (-70)}{3} = \frac{70}{3} \approx 23.33 \][/tex]
Since [tex]\( 23.33 \)[/tex] (rate of increase of [tex]\( g \)[/tex]) is greater than [tex]\( 18.67 \)[/tex] (rate of increase of [tex]\( f \)[/tex]), [tex]\( g \)[/tex] has a faster average rate of increase than [tex]\( f \)[/tex].
### Conclusion
Based on the provided analysis:
- Both [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are increasing.
- [tex]\( g \)[/tex] increases at a faster average rate than [tex]\( f \)[/tex].
Thus, the correct statement is:
[tex]\[ \boxed{\text{C. Both functions are increasing, but function } g \text{ increases at a faster average rate.}} \][/tex]
