To find the value of [tex]\( f(x) \)[/tex] for [tex]\( f(x) = \frac{8}{1 + 3 e^{-0.7 x}} \)[/tex] when [tex]\( x = -1 \)[/tex], we need to follow these steps:
1. Substitute [tex]\( x = -1 \)[/tex] into the function:
[tex]\[
f(-1) = \frac{8}{1 + 3 e^{-0.7 \cdot (-1)}}
\][/tex]
2. Simplify the exponent:
[tex]\[
-0.7 \cdot (-1) = 0.7
\][/tex]
3. Rewrite the function with the simplified exponent:
[tex]\[
f(-1) = \frac{8}{1 + 3 e^{0.7}}
\][/tex]
4. Calculate [tex]\( e^{0.7} \)[/tex], this is approximately:
[tex]\[
e^{0.7} \approx 2.01375
\][/tex]
5. Substitute this value back into the denominator:
[tex]\[
f(-1) = \frac{8}{1 + 3 \cdot 2.01375}
\][/tex]
6. Calculate the value inside the denominator:
[tex]\[
3 \cdot 2.01375 = 6.04125
\][/tex]
[tex]\[
1 + 6.04125 = 7.04125
\][/tex]
7. Divide the numerator by this result:
[tex]\[
f(-1) = \frac{8}{7.04125} \approx 1.1361605924567681
\][/tex]
8. Round this result to the nearest hundredth:
[tex]\[
1.1361605924567681 \approx 1.14
\][/tex]
Thus, the value of [tex]\( f(-1) \)[/tex] rounded to the nearest hundredth is [tex]\( 1.14 \)[/tex].
