The range of the function $f(x) = -2(2^x) + 5$ decreases over its domain.

Explanation:

• The function is an exponential function that has been reflected vertically (due to the negative sign) and shifted upwards by 5 units.
• As $x$ increases, the term $2^x$ grows larger, and since it is multiplied by -2, the overall function value $f(x)$ decreases.
• As $x$ decreases, the term $2^x$ approaches 0, making the function approach the horizontal asymptote $y = 5$ from below.

So, across the domain $(-\infty, \infty)$:

• For decreasing $x$: $f(x)$ approaches 5 from below.
• For increasing $x$: $f(x)$ decreases towards negative infinity.

Thus, the function's range decreases as you move from left to right along the domain.

Would you like further clarification on this, or any other details?

Related Questions:

1. How does the reflection affect the function's behavior?
2. What would the range be if the function weren't reflected (i.e., without the negative sign)?
3. Can you explain how the base of the exponential function affects its growth?
4. How do vertical shifts change the range of exponential functions?
5. What other transformations could affect the range and direction of an exponential function?

Tip: When analyzing the range of a function, always pay attention to both reflections (sign changes) and vertical shifts. These transformations directly impact how the function behaves over its domain.