Graph the function. Then identify the domain, range, and \( y \)-intercept, and state whether the function is increasing or decreasing. \( f(x)=e^{-4 x} \) Identify the function's y-intercept. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function's y-intercept is \( (0,1) \). (Type an ordered pair.) B. The function has no \( y \)-intercept. The function is decreasing.

To graph the function \( f(x) = e^{-4x} \), we can use a graphing calculator or software. However, since we are not provided with a specific graphing tool, we can analyze the function's properties to determine the domain, range, and \( y \)-intercept. 1. **Domain:** The domain of a function is the set of all possible input values for which the function is defined. For the function \( f(x) = e^{-4x} \), the domain is all real numbers since the exponential function is defined for all real numbers. 2. **Range:** The range of a function is the set of all possible output values. For the function \( f(x) = e^{-4x} \), the range is all positive real numbers since the exponential function with a negative exponent is always positive. 3. **\( y \)-intercept:** The \( y \)-intercept is the point where the function crosses the \( y \)-axis, i.e., when \( x = 0 \). To find the \( y \)-intercept, we substitute \( x = 0 \) into the function: \[ f(0) = e^{-4(0)} = e^0 = 1 \] Therefore, the \( y \)-intercept is \( (0, 1) \). 4. **Increasing or Decreasing:** To determine if the function is increasing or decreasing, we can analyze the derivative of the function. The derivative of \( f(x) = e^{-4x} \) is: \[ f'(x) = -4e^{-4x} \] Since the derivative is negative for all \( x \), the function is decreasing. Therefore, the correct choice is: A. The function's \( y \)-intercept is \( (0, 1) \).