To find the domain of the given function $f(x) = 2^{\frac{3}{4-3x}}$, we need to ensure that:

1. The base of the exponent (2) is positive, so it will not introduce any restrictions.
2. The denominator in the exponent, $4 - 3x$, must not be zero because division by zero is undefined.

Steps:

1. Solve for when $4 - 3x = 0$: $4 - 3x = 0 \quad \Rightarrow \quad x = \frac{4}{3}.$

2. Therefore, the function is undefined at $x = \frac{4}{3}$, and the domain excludes this value.

Domain in Interval Notation:

The domain is all real numbers except $x = \frac{4}{3}$. In interval notation: $(-\infty, \frac{4}{3}) \cup (\frac{4}{3}, \infty).$

Let me know if you need further explanation or have any questions.

Related Questions:

1. How do you determine the domain of a logarithmic function?
2. How do discontinuities affect the domain of a function?
3. What are the steps to simplify fractional exponents in functions?
4. How does $a^{f(x)}$ behave as $x \to \infty$ or $x \to -\infty$?
5. Can you graphically visualize $f(x) = 2^{\frac{3}{4-3x}}$ to see the undefined points?

Tip:

Always check for division by zero and negative values in square roots