To find the [tex]\( y \)[/tex]-intercept of the graph of the function [tex]\( f(x) = \frac{15}{1 + 4 e^{-0.2 x}} \)[/tex], we need to evaluate the function [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex].
1. Start by substituting [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = \frac{15}{1 + 4 e^{-0.2 \cdot 0}} \][/tex]
2. Simplify the exponent:
[tex]\[ e^{-0.2 \cdot 0} = e^{0} = 1 \][/tex]
3. Substitute [tex]\( e^{0} = 1 \)[/tex] back into the equation:
[tex]\[ f(0) = \frac{15}{1 + 4 \cdot 1} \][/tex]
4. Simplify the denominator:
[tex]\[ 1 + 4 \cdot 1 = 1 + 4 = 5 \][/tex]
5. Finish simplifying the function:
[tex]\[ f(0) = \frac{15}{5} = 3 \][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept of the graph of [tex]\( f(x) \)[/tex] is:
[tex]\[ y \text{-intercept} = 3 \][/tex]
