Answer :
To determine which points represent ordered pairs on the function [tex]\( g(x) \)[/tex] which is the reflection of the function [tex]\( f(x) = -\frac{2}{7} \left(\frac{5}{3}\right)^x \)[/tex] over the y-axis, follow these steps:
1. Define the Reflection:
The reflection of [tex]\( f(x) \)[/tex] over the y-axis changes the input [tex]\( x \)[/tex] to [tex]\( -x \)[/tex]. Therefore, [tex]\( g(x) = f(-x) \)[/tex].
2. Substitute [tex]\( -x \)[/tex] into [tex]\( f(x) \)[/tex]:
Calculate the form of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = f(-x) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-x} \][/tex]
3. Evaluate Potential Points:
We will check each given point [tex]\((x, y)\)[/tex] to see if it lies on the function [tex]\( g(x) = f(-x) \)[/tex], i.e., if [tex]\( y = g(x) \)[/tex].
- Point [tex]\((-7, -10.206)\)[/tex]:
[tex]\[ g(-7) = f(7) = -\frac{2}{7} \left(\frac{5}{3}\right)^7 \][/tex]
When this is calculated, the result matches approximately [tex]\(-10.206\)[/tex], so this point is on [tex]\( g(x) \)[/tex].
- Point [tex]\((-2.5, -4.474)\)[/tex]:
[tex]\[ g(-2.5) = f(2.5) = -\frac{2}{7} \left(\frac{5}{3}\right)^{2.5} \][/tex]
When this is calculated, the result does not match [tex]\(-4.474\)[/tex], so this point is not on [tex]\( g(x) \)[/tex].
- Point [tex]\((0, -1.773)\)[/tex]:
[tex]\[ g(0) = f(0) = -\frac{2}{7} \left(\frac{5}{3}\right)^0 = -\frac{2}{7} \][/tex]
This results in [tex]\(-\frac{2}{7}\)[/tex] which is [tex]\(\approx -0.286\)[/tex], not [tex]\(-1.773\)[/tex]. So, this point does not lie on [tex]\( g(x) \)[/tex].
- Point [tex]\((0.5, -0.221)\)[/tex]:
[tex]\[ g(0.5) = f(-0.5) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-0.5} \][/tex]
When this is calculated, the result matches approximately [tex]\(-0.221\)[/tex], so this point is on [tex]\( g(x) \)[/tex].
- Point [tex]\((4, -0.017)\)[/tex]:
[tex]\[ g(4) = f(-4) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-4} \][/tex]
When this is calculated, the result does not match [tex]\(-0.017\)[/tex], so this point is not on [tex]\( g(x) \)[/tex].
- Point [tex]\((9, -0.003)\)[/tex]:
[tex]\[ g(9) = f(-9) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-9} \][/tex]
When this is calculated, the result matches approximately [tex]\(-0.003\)[/tex], so this point is on [tex]\( g(x) \)[/tex].
4. Conclusion:
Based on the evaluations, the points that represent ordered pairs on [tex]\( g(x) \)[/tex] are:
[tex]\[ (-7, -10.206), \quad (0.5, -0.221), \quad (9, -0.003) \][/tex]
Therefore, the points [tex]\((-7, -10.206)\)[/tex], [tex]\((0.5, -0.221)\)[/tex], and [tex]\((9, -0.003)\)[/tex] are the points that lie on [tex]\( g(x) \)[/tex].
1. Define the Reflection:
The reflection of [tex]\( f(x) \)[/tex] over the y-axis changes the input [tex]\( x \)[/tex] to [tex]\( -x \)[/tex]. Therefore, [tex]\( g(x) = f(-x) \)[/tex].
2. Substitute [tex]\( -x \)[/tex] into [tex]\( f(x) \)[/tex]:
Calculate the form of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = f(-x) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-x} \][/tex]
3. Evaluate Potential Points:
We will check each given point [tex]\((x, y)\)[/tex] to see if it lies on the function [tex]\( g(x) = f(-x) \)[/tex], i.e., if [tex]\( y = g(x) \)[/tex].
- Point [tex]\((-7, -10.206)\)[/tex]:
[tex]\[ g(-7) = f(7) = -\frac{2}{7} \left(\frac{5}{3}\right)^7 \][/tex]
When this is calculated, the result matches approximately [tex]\(-10.206\)[/tex], so this point is on [tex]\( g(x) \)[/tex].
- Point [tex]\((-2.5, -4.474)\)[/tex]:
[tex]\[ g(-2.5) = f(2.5) = -\frac{2}{7} \left(\frac{5}{3}\right)^{2.5} \][/tex]
When this is calculated, the result does not match [tex]\(-4.474\)[/tex], so this point is not on [tex]\( g(x) \)[/tex].
- Point [tex]\((0, -1.773)\)[/tex]:
[tex]\[ g(0) = f(0) = -\frac{2}{7} \left(\frac{5}{3}\right)^0 = -\frac{2}{7} \][/tex]
This results in [tex]\(-\frac{2}{7}\)[/tex] which is [tex]\(\approx -0.286\)[/tex], not [tex]\(-1.773\)[/tex]. So, this point does not lie on [tex]\( g(x) \)[/tex].
- Point [tex]\((0.5, -0.221)\)[/tex]:
[tex]\[ g(0.5) = f(-0.5) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-0.5} \][/tex]
When this is calculated, the result matches approximately [tex]\(-0.221\)[/tex], so this point is on [tex]\( g(x) \)[/tex].
- Point [tex]\((4, -0.017)\)[/tex]:
[tex]\[ g(4) = f(-4) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-4} \][/tex]
When this is calculated, the result does not match [tex]\(-0.017\)[/tex], so this point is not on [tex]\( g(x) \)[/tex].
- Point [tex]\((9, -0.003)\)[/tex]:
[tex]\[ g(9) = f(-9) = -\frac{2}{7} \left(\frac{5}{3}\right)^{-9} \][/tex]
When this is calculated, the result matches approximately [tex]\(-0.003\)[/tex], so this point is on [tex]\( g(x) \)[/tex].
4. Conclusion:
Based on the evaluations, the points that represent ordered pairs on [tex]\( g(x) \)[/tex] are:
[tex]\[ (-7, -10.206), \quad (0.5, -0.221), \quad (9, -0.003) \][/tex]
Therefore, the points [tex]\((-7, -10.206)\)[/tex], [tex]\((0.5, -0.221)\)[/tex], and [tex]\((9, -0.003)\)[/tex] are the points that lie on [tex]\( g(x) \)[/tex].
