Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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].
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
