To solve the given exponential function
[tex]\[
y = -2 \left( \frac{1}{3} \right)^x + 6
\][/tex]
we need to find the values of [tex]\( y \)[/tex] for specified values of [tex]\( x \)[/tex]. Let's evaluate this function step-by-step for [tex]\( x = 1 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = 3 \)[/tex].
1. For [tex]\( x = 1 \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[
y = -2 \left( \frac{1}{3} \right)^1 + 6
\][/tex]
Simplify the power:
[tex]\[
\left( \frac{1}{3} \right)^1 = \frac{1}{3}
\][/tex]
So the equation becomes:
[tex]\[
y = -2 \times \frac{1}{3} + 6
\][/tex]
Multiply:
[tex]\[
y = -\frac{2}{3} + 6
\][/tex]
Convert 6 to a fraction with the same denominator:
[tex]\[
y = -\frac{2}{3} + \frac{18}{3}
\][/tex]
Combine the fractions:
[tex]\[
y = \frac{-2 + 18}{3} = \frac{16}{3}
\][/tex]
Hence, the value of [tex]\( y \)[/tex] when [tex]\( x = 1 \)[/tex] is approximately:
[tex]\[
y \approx 5.333
\][/tex]
2. For [tex]\( x = 2 \)[/tex]:
Substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[
y = -2 \left( \frac{1}{3} \right)^2 + 6
\][/tex]
Simplify the power:
[tex]\[
\left( \frac{1}{3} \right)^2 = \frac{1}{9}
\][/tex]
So the equation becomes:
[tex]\[
y = -2 \times \frac{1}{9} + 6
\][/tex]
Multiply:
[tex]\[
y = -\frac{2}{9} + 6
\][/tex]
Convert 6 to a fraction with the same denominator:
[tex]\[
y = -\frac{2}{9} + \frac{54}{9}
\][/tex]
Combine the fractions:
[tex]\[
y = \frac{-2 + 54}{9} = \frac{52}{9}
\][/tex]
Hence, the value of [tex]\( y \)[/tex] when [tex]\( x = 2 \)[/tex] is approximately:
[tex]\[
y \approx 5.778
\][/tex]
3. For [tex]\( x = 3 \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the function:
[tex]\[
y = -2 \left( \frac{1}{3} \right)^3 + 6
\][/tex]
Simplify the power:
[tex]\[
\left( \frac{1}{3} \right)^3 = \frac{1}{27}
\][/tex]
So the equation becomes:
[tex]\[
y = -2 \times \frac{1}{27} + 6
\][/tex]
Multiply:
[tex]\[
y = -\frac{2}{27} + 6
\][/tex]
Convert 6 to a fraction with the same denominator:
[tex]\[
y = -\frac{2}{27} + \frac{162}{27}
\][/tex]
Combine the fractions:
[tex]\[
y = \frac{-2 + 162}{27} = \frac{160}{27}
\][/tex]
Hence, the value of [tex]\( y \)[/tex] when [tex]\( x = 3 \)[/tex] is approximately:
[tex]\[
y \approx 5.926
\][/tex]
So, the calculated values of [tex]\( y \)[/tex] are:
- For [tex]\( x = 1 \)[/tex], [tex]\( y \approx 5.333 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y \approx 5.778 \)[/tex]
- For [tex]\( x = 3 \)[/tex], [tex]\( y \approx 5.926 \)[/tex]
