Finding Domain and Range From the Graph of an Exponential Function ... - Free Printable

Let's carefully analyze the given exponential function and its graph to find the domain and range.

---

Given Function:

$$
f(x) = -\frac{1}{2} \cdot 3^x
$$

This is an exponential function with:
- Base: $3$ (which is greater than 1),
- Coefficient: $-\frac{1}{2}$, which reflects the graph over the x-axis and vertically compresses it.

---

Step 1: Understand the Graph

From the graph:
- The curve starts from the left (as $x \to -\infty$), approaching $y = 0$ from above.
- As $x$ increases, the function decreases rapidly toward negative infinity.
- The graph never touches or crosses the x-axis; it approaches $y = 0$ as a horizontal asymptote.
- The function is defined for all real numbers.

---

Step 2: Find the Domain

The domain is the set of all possible input values ($x$-values) for which the function is defined.

Since $3^x$ is defined for all real numbers, and multiplying by a constant doesn't change that, the function $f(x) = -\frac{1}{2} \cdot 3^x$ is defined for all real numbers.

✔ Domain:
$$
(-\infty, \infty) \quad \text{or} \quad \mathbb{R}
$$

---

Step 3: Find the Range

The range is the set of all possible output values ($y$-values).

Let’s analyze the behavior of $f(x)$:

- As $x \to -\infty$:
$3^x \to 0$, so $f(x) = -\frac{1}{2} \cdot 3^x \to 0$ from the negative side? Wait — let's check:

Actually:
- $3^x \to 0^+$ as $x \to -\infty$
- So $-\frac{1}{2} \cdot 3^x \to 0^-$ → approaches 0 from below (i.e., negative values close to zero)

So:
- As $x \to -\infty$, $f(x) \to 0^-$
- As $x \to \infty$, $3^x \to \infty$, so $f(x) = -\frac{1}{2} \cdot 3^x \to -\infty$

Therefore, the function takes on all negative real values, but never reaches 0 or any positive value.

✔ Range:
$$
(-\infty, 0)
$$

Note: It approaches 0 but never reaches it. So 0 is not included.

---

✔ Final Answer:

- Domain: $(-\infty, \infty)$ or $\mathbb{R}$ — all real numbers
- Range: $(-\infty, 0)$ — all negative real numbers

---

🔍 Why is this correct?

- Exponential functions like $a^x$ have domain $\mathbb{R}$.
- Multiplying by a negative coefficient flips the graph over the x-axis.
- Since $3^x > 0$ for all $x$, then $-\frac{1}{2} \cdot 3^x < 0$ for all $x$, so outputs are always negative.
- No upper bound (approaches 0), but no lower bound (goes to $-\infty$).

---

📌 Summary:

| Feature | Value |
|------------|-------|
| Domain | $(-\infty, \infty)$ |
| Range | $(-\infty, 0)$ |

✔ This matches the handwritten answer in the image.

Let me know if you'd like to see how to sketch this graph step-by-step!