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Answer :
To determine if the point [tex]\((-2, \frac{1}{25})\)[/tex] is on the graph of the function [tex]\(g(x) = \left(\frac{1}{5}\right)^x\)[/tex], we can follow these steps:
1. Substitute the x-value into the function:
First, substitute [tex]\(x = -2\)[/tex] into the function [tex]\(g(x)\)[/tex]. This means you'll calculate [tex]\(g(-2)\)[/tex].
2. Calculate the function value:
[tex]\[
g(-2) = \left(\frac{1}{5}\right)^{-2}
\][/tex]
Recall the rule for negative exponents: [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. So, we have:
[tex]\[
g(-2) = \left(\frac{5}{1}\right)^{2} = 5^2
\][/tex]
[tex]\[
g(-2) = 25
\][/tex]
3. Compare the calculated value with the expected y-value:
Check if [tex]\(g(-2)\)[/tex] equals [tex]\(\frac{1}{25}\)[/tex]:
- The calculated value of [tex]\(g(-2)\)[/tex] is 25.
- The given y-coordinate is [tex]\(\frac{1}{25}\)[/tex].
Since 25 does not equal [tex]\(\frac{1}{25}\)[/tex], the point [tex]\((-2, \frac{1}{25})\)[/tex] is not on the graph of the function [tex]\(g(x) = \left(\frac{1}{5}\right)^x\)[/tex].
Therefore, the conclusion is that the point [tex]\((-2, \frac{1}{25})\)[/tex] is not on the graph.
1. Substitute the x-value into the function:
First, substitute [tex]\(x = -2\)[/tex] into the function [tex]\(g(x)\)[/tex]. This means you'll calculate [tex]\(g(-2)\)[/tex].
2. Calculate the function value:
[tex]\[
g(-2) = \left(\frac{1}{5}\right)^{-2}
\][/tex]
Recall the rule for negative exponents: [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. So, we have:
[tex]\[
g(-2) = \left(\frac{5}{1}\right)^{2} = 5^2
\][/tex]
[tex]\[
g(-2) = 25
\][/tex]
3. Compare the calculated value with the expected y-value:
Check if [tex]\(g(-2)\)[/tex] equals [tex]\(\frac{1}{25}\)[/tex]:
- The calculated value of [tex]\(g(-2)\)[/tex] is 25.
- The given y-coordinate is [tex]\(\frac{1}{25}\)[/tex].
Since 25 does not equal [tex]\(\frac{1}{25}\)[/tex], the point [tex]\((-2, \frac{1}{25})\)[/tex] is not on the graph of the function [tex]\(g(x) = \left(\frac{1}{5}\right)^x\)[/tex].
Therefore, the conclusion is that the point [tex]\((-2, \frac{1}{25})\)[/tex] is not on the graph.
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