Question: What is the range of this exponential function?
y = -3 \cdot 4^x
{y | y < 0}
all real numbers
{y | y > 0}
{y | y \geq 0}
{y | y \leq 0}
Graph shape: exponential function y = -3\cdot 4^x; the curve stays below the x-axis, approaches 0 from below as x increases, and decreases without bound as x decreases.
1. **State the problem:** Find the range of the exponential function $$y = -3 \cdot 4^x$$.
2. **Recall the properties of exponential functions:**
- The base $$4^x$$ is always positive for all real $$x$$, so $$4^x > 0$$.
- Multiplying by $$-3$$ flips the graph vertically and scales it.
3. **Analyze the function:**
- Since $$4^x > 0$$, then $$-3 \cdot 4^x < 0$$ because of the negative multiplier.
- As $$x \to \infty$$, $$4^x \to \infty$$, so $$y = -3 \cdot 4^x \to -\infty$$.
- As $$x \to -\infty$$, $$4^x \to 0^+$$, so $$y = -3 \cdot 4^x \to 0^-$$ (approaches zero from below).
4. **Determine the range:**
- The function values are always negative and approach zero but never reach zero.
- Therefore, the range is all $$y$$ such that $$y < 0$$.
**Final answer:**
$$\boxed{\{y \mid y < 0\}}$$