What is the \( y \)-intercept of \( f(x)=e^{-x}+3 \) ? \( \square \) Type an integer or a simplified fraction.)

To find the \( y \)-intercept of the function \( f(x) = e^{-x} + 3 \), we evaluate the function at \( x = 0 \). Substituting \( 0 \) into the equation gives us \( f(0) = e^{0} + 3 = 1 + 3 = 4 \). Therefore, the \( y \)-intercept is \( 4 \). The \( y \)-intercept is a critical point for understanding a function’s behavior. It often represents the starting point when analyzing functions in graphs or real-world applications, such as modeling populations or tracking financial growth. In these scenarios, the \( y \)-intercept can indicate initial conditions or baseline values. When calculating \( y \)-intercepts, a common mistake is forgetting to evaluate the function at \( x = 0 \) or misinterpreting the outputs of exponential functions, especially when using negative exponents. Remembering the properties of exponents is key: for \( e \), any negative exponent will yield a value between \( 0 \) and \( 1 \) before any addition, so make sure to assess each component carefully!