https://file.aiquickdraw.com/mathbot/file/172579864840291ajtvov.jpg

The function given in the image is:

\[
f(x) = (x + 2)^{\frac{3}{4}}
\]

This function involves raising \(x + 2\) to a fractional exponent, specifically \( \frac{3}{4} \). 

To determine the domain of this function (i.e., the values of \(x\) for which the function is defined), we need to consider the properties of fractional exponents. 

- When the exponent's denominator is even (like 4 here), the base must be non-negative, because raising a negative number to a fractional exponent with an even denominator results in a complex number, which is outside the scope of real-valued functions.

Thus, for the function \(f(x)\) to be defined for real numbers, the expression inside the parentheses must be non-negative:

\[
x + 2 \geq 0
\]

Solving this inequality:

\[
x \geq -2
\]

Therefore, the domain of the function is \(x \in [-2, \infty)\).

This corresponds to option **B** in the question.

Would you like further details or explanations?

Here are some follow-up questions you might find interesting:
1. What is the range of the function \( f(x) = (x + 2)^{\frac{3}{4}} \)?
2. How do fractional exponents behave differently when the denominator is even vs. odd?
3. What happens to the graph of the function \( f(x) \) as \( x \to \infty \)?
4. How can you transform the function \( f(x) = (x + 2)^{\frac{3}{4}} \) to shift or stretch its graph?
5. If the exponent were \( \frac{5}{4} \) instead, how would the domain change?

**Tip:** When dealing with fractional exponents, always check if the base can be negative for real-valued functions, especially if the denominator of the exponent is even!

For the function f(x) = (x + 2)^(3/4), what are all the values of x for which the function is defined?

Explore the domain of the function f(x) = (x + 2)^(3/4), where fractional exponents and algebraic inequalities determine the valid values of x. This problem focuses on algebraic concepts and is suitable for high school students.

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