Which points are on the graph of [tex][tex]$k(x)=\left(\frac{1}{4}\right)^x$[/tex]?[/tex]

Select each correct answer.

A. [tex]\left(3, \frac{1}{64}\right)[/tex]
B. [tex](-1, 4)[/tex]
C. [tex](1, 4)[/tex]
D. [tex]\left(0, \frac{1}{4}\right)[/tex]



Answer :

To determine which points lie on the graph of the function [tex]\( k(x) = \left( \frac{1}{4} \right)^x \)[/tex], let's evaluate [tex]\( k(x) \)[/tex] at the given [tex]\( x \)[/tex]-values and compare with the corresponding [tex]\( y \)[/tex]-values in each point:

1. Point [tex]\(\left( 3, \frac{1}{64} \right)\)[/tex]:
- Evaluate [tex]\( k(3) \)[/tex]:
[tex]\[ k(3) = \left( \frac{1}{4} \right)^3 = \frac{1}{4^3} = \frac{1}{64} \][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(\frac{1}{64}\)[/tex]. Thus, [tex]\(\left( 3, \frac{1}{64} \right)\)[/tex] is on the graph.

2. Point [tex]\((-1, 4)\)[/tex]:
- Evaluate [tex]\( k(-1) \)[/tex]:
[tex]\[ k(-1) = \left( \frac{1}{4} \right)^{-1} = \left( \frac{4}{1} \right) = 4 \][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(4\)[/tex]. Thus, [tex]\((-1, 4)\)[/tex] is on the graph.

3. Point [tex]\((1, 4)\)[/tex]:
- Evaluate [tex]\( k(1) \)[/tex]:
[tex]\[ k(1) = \left( \frac{1}{4} \right)^1 = \frac{1}{4} \][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(\frac{1}{4}\)[/tex], which is not equal to [tex]\(4\)[/tex]. Thus, [tex]\((1, 4)\)[/tex] is not on the graph.

4. Point [tex]\(\left( 0, \frac{1}{4} \right)\)[/tex]:
- Evaluate [tex]\( k(0) \)[/tex]:
[tex]\[ k(0) = \left( \frac{1}{4} \right)^0 = 1 \][/tex]
- The [tex]\( y \)[/tex]-value we get is [tex]\(1\)[/tex], which is not equal to [tex]\(\frac{1}{4}\)[/tex]. Thus, [tex]\(\left( 0, \frac{1}{4} \right)\)[/tex] is not on the graph.

Therefore, the points that are on the graph of [tex]\( k(x) = \left( \frac{1}{4} \right)^x \)[/tex] are:
[tex]\[ \left( 3, \frac{1}{64} \right) \quad \text{and} \quad (-1, 4) \][/tex]