To find the value of the function [tex]\( f(x) = \frac{8}{1 + 3 e^{-0.7 x}} \)[/tex] at [tex]\( x = -3 \)[/tex], follow these steps:
1. Substitute [tex]\( x = -3 \)[/tex] into the function:
[tex]\[
f(-3) = \frac{8}{1 + 3 e^{-0.7 (-3)}}
\][/tex]
2. Simplify the exponent:
[tex]\[
-0.7 \times (-3) = 2.1
\][/tex]
Therefore,
[tex]\[
f(-3) = \frac{8}{1 + 3 e^{2.1}}
\][/tex]
3. Calculate [tex]\( e^{2.1} \)[/tex]:
The value of the exponential function [tex]\( e^{2.1} \approx 8.1662 \)[/tex].
4. Substitute [tex]\( e^{2.1} \)[/tex] back into the function:
[tex]\[
f(-3) = \frac{8}{1 + 3 \times 8.1662}
\][/tex]
5. Multiply and add:
[tex]\[
3 \times 8.1662 = 24.4986
\][/tex]
[tex]\[
1 + 24.4986 = 25.4986
\][/tex]
6. Divide:
[tex]\[
f(-3) = \frac{8}{25.4986} \approx 0.31374382590567385
\][/tex]
7. Round the result to the nearest hundredth:
[tex]\[
f(-3) \approx 0.31
\][/tex]
Therefore, the value of [tex]\( f(-3) \)[/tex] rounded to the nearest hundredth is:
[tex]\[
0.31
\][/tex]
