https://file.aiquickdraw.com/mathbot/file/1729005860715ewz3gxgp.jpg

Let's analyze the function given: \( f(x) = 4^{x^2} \).

### Key observations:
- \( x^2 \) is always non-negative, meaning \( x^2 \geq 0 \) for all real numbers \( x \).
- Therefore, the smallest value that \( 4^{x^2} \) can take is \( 4^0 = 1 \), which occurs when \( x = 0 \).
- As \( x^2 \) increases, \( 4^{x^2} \) will increase rapidly. For example, \( f(1) = 4^1 = 4 \), and for larger \( x \), the function will take values greater than 4.
- The function approaches \( 1 \) as \( x \to 0 \), but it never becomes less than 1.

### Conclusion:
The range of \( f(x) = 4^{x^2} \) is all real numbers **greater than or equal to 1**. However, this option is not explicitly listed among the choices.

Given the available options, **none of them are precisely correct**. However, based on the closest choice, **D. All real numbers greater than or equal to 4** is incorrect since the minimum value of the function is 1, not 4. **None of the listed options** seem accurate.

Would you like further details, or do you have any questions?

---

Here are 5 questions related to this topic:
1. How would the range change if the base were less than 4, for example, \( f(x) = 2^{x^2} \)?
2. What happens to the graph of \( f(x) = 4^{x^2} \) as \( x \) becomes very large or very small?
3. How do transformations like shifting or stretching affect the range of this function?
4. Could this function ever take a negative value? Why or why not?
5. How does the range differ between exponential functions like \( f(x) = 4^x \) and \( f(x) = 4^{x^2} \)?

**Tip:** When working with functions involving exponents, always consider how the exponent behaves over all possible inputs, especially for even powers or absolute values.

Understand the range of the exponential function f(x) = 4^{x^2}. Learn how the function behaves for different values of x, the impact of squaring the exponent, and why the minimum value is 1. Suitable for high school students studying exponential and quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation