1. **State the problem:** We want to find which of the numbers 4, -0.5, 5, and 1 are included in the set of values of the function $$y = 16^{-x} + 4$$. 2. **Recall the function and its properties:** The function is $$y = 16^{-x} + 4$$. Since $$16 = 2^4$$, we can rewrite it as $$y = (2^4)^{-x} + 4 = 2^{-4x} + 4$$. 3. **Analyze the range of the function:** The term $$2^{-4x}$$ is always positive because an exponential function with a positive base is always positive. 4. **Minimum value of $$y$$:** Since $$2^{-4x} > 0$$ for all real $$x$$, the smallest value of $$y$$ is when $$2^{-4x} \to 0$$, which happens as $$x \to +\infty$$. Thus, $$y_{min} = 4$$. 5. **Maximum value of $$y$$:** As $$x \to -\infty$$, $$2^{-4x} = 2^{4|x|} \to +\infty$$, so $$y \to +\infty$$. 6. **Range conclusion:** The range of $$y$$ is $$[4, +\infty)$$. 7. **Check each number:** - a) 4: Included because $$y_{min} = 4$$. - b) -0.5: Not included because $$y$$ is always $$\geq 4$$. - c) 5: Included because $$5 > 4$$ and $$y$$ can take any value greater than or equal to 4. - d) 1: Not included because $$1 < 4$$. **Final answer:** The set of values includes 4 and 5.