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}
Submit
Work it out
Graph shape: no graph is shown; the math content is an exponential function $y = -3 \cdot 4^x$, which is always negative, approaching 0 from below.
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 sign and scales the function.
3. **Analyze the function:**
- Since $$4^x > 0$$, multiplying by $-3$ gives $$y = -3 \cdot 4^x < 0$$ for all $x$.
- The function approaches zero from below as $x \to -\infty$ because $$4^x \to 0^+$$, so $$y \to 0^-$$.
- The function decreases without bound as $x \to \infty$ because $$4^x \to \infty$$, so $$y \to -\infty$$.
4. **Conclusion:**
- The range of $$y = -3 \cdot 4^x$$ is all negative real numbers.
- In set notation, the range is $$\{ y \mid y < 0 \}$$.
**Final answer:**
$$\boxed{\{ y \mid y < 0 \}}$$