To analyze the function $f(x) = 4\left(\frac{3}{2}\right)^x$, let's determine its domain and range.

1. Domain

The domain of this function is all the values of $x$ for which the function is defined. Since $\left(\frac{3}{2}\right)^x$ is defined for all real numbers $x$, there are no restrictions.

• Domain: $(-\infty, \infty)$.

2. Range

The function $f(x) = 4\left(\frac{3}{2}\right)^x$ represents an exponential function. Key properties of exponential functions include:

• The base $\frac{3}{2} > 1$, so the function is increasing.
• Exponential functions never produce zero or negative values but approach $0$ asymptotically as $x \to -\infty$.

Here, the constant $4$ scales the range. Therefore:

• $f(x) > 0$ for all $x$.
• Range: $(0, \infty)$.

Final Answer:

• Domain: $(-\infty, \infty)$
• Range: $(0, \infty)$

Would you like a graphical explanation or a step-by-step process? Let me know!

5 Related Questions:

1. How does changing the base $\frac{3}{2}$ affect the function's growth?
2. What happens to the range if the constant $4$ changes to a negative value?
3. How would you find the horizontal asymptote of this function?
4. Can you calculate the value of $f(x)$ at specific points like $x = 0$, $x = 1$, and $x = -1$?
5. How would you determine where $f(x) > 10$?

Tip:

For exponential functions $f(x) = a b^x$, the base $b > 1$ indicates growth, while $0 < b < 1$ indicates decay.