Answer :
To determine which graph represents the function [tex]\( f(x) = -3^x - 2 \)[/tex], we can evaluate the function at several values of [tex]\( x \)[/tex]. Let's see the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -2, -1, 0, 1, 2 \)[/tex]:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -3^{-2} - 2 = -\frac{1}{3^2} - 2 = -\frac{1}{9} - 2 \approx -2.111 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -3^{-1} - 2 = -\frac{1}{3} - 2 \approx -2.333 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -3^0 - 2 = -1 - 2 = -3 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -3^1 - 2 = -3 - 2 = -5 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -3^2 - 2 = -9 - 2 = -11 \][/tex]
Summarizing, we have the following [tex]\( (x, f(x)) \)[/tex] pairs:
[tex]\[ (-2, -2.111), (-1, -2.333), (0, -3), (1, -5), (2, -11) \][/tex]
To match this function to a graph, check the graph that follows these key characteristics:
- The function is decreasing since larger [tex]\( x \)[/tex] values produce significantly smaller (more negative) [tex]\( f(x) \)[/tex] values.
- The function includes the points: [tex]\((-2, -2.111)\)[/tex], [tex]\((-1, -2.333)\)[/tex], [tex]\((0, -3)\)[/tex], [tex]\((1, -5)\)[/tex], and [tex]\((2, -11)\)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] rapidly drops below [tex]\(-2\)[/tex] (i.e., becomes more negative).
Look for a graph that corroborates these points and trends.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = -3^{-2} - 2 = -\frac{1}{3^2} - 2 = -\frac{1}{9} - 2 \approx -2.111 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -3^{-1} - 2 = -\frac{1}{3} - 2 \approx -2.333 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -3^0 - 2 = -1 - 2 = -3 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -3^1 - 2 = -3 - 2 = -5 \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = -3^2 - 2 = -9 - 2 = -11 \][/tex]
Summarizing, we have the following [tex]\( (x, f(x)) \)[/tex] pairs:
[tex]\[ (-2, -2.111), (-1, -2.333), (0, -3), (1, -5), (2, -11) \][/tex]
To match this function to a graph, check the graph that follows these key characteristics:
- The function is decreasing since larger [tex]\( x \)[/tex] values produce significantly smaller (more negative) [tex]\( f(x) \)[/tex] values.
- The function includes the points: [tex]\((-2, -2.111)\)[/tex], [tex]\((-1, -2.333)\)[/tex], [tex]\((0, -3)\)[/tex], [tex]\((1, -5)\)[/tex], and [tex]\((2, -11)\)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( f(x) \)[/tex] rapidly drops below [tex]\(-2\)[/tex] (i.e., becomes more negative).
Look for a graph that corroborates these points and trends.
