https://file.aiquickdraw.com/mathbot/file/1725475995039b79ks9gj.jpg

To determine the range of the function \( f(x) = x^3 + 2 \), let's analyze it step by step.

### Step 1: Consider the function
The function is given as \( f(x) = x^3 + 2 \).

### Step 2: Analyze the cubic term
The term \( x^3 \) is a cubic function, and cubic functions have the following properties:
- They are continuous.
- They are defined for all real numbers \( x \).
- As \( x \) approaches \( \infty \), \( x^3 \) approaches \( \infty \).
- As \( x \) approaches \( -\infty \), \( x^3 \) approaches \( -\infty \).

### Step 3: Determine the range
Since \( x^3 \) can take any real value, \( x^3 + 2 \) will also take any real value, but shifted upwards by 2 units. 

Thus, the range of \( f(x) = x^3 + 2 \) is all real numbers.

### Step 4: Select the correct answer
The correct answer from the options provided is **"All real numbers"**.

Would you like further details or have any questions?

Here are five related questions to expand your understanding:
1. How do you find the domain of a function?
2. What is the difference between the range and the codomain of a function?
3. Can you explain why a cubic function's range is all real numbers?
4. How does shifting a function vertically affect its range?
5. What is the impact of adding a constant to a function on its graph?

**Tip:** Always start analyzing the range by considering the behavior of the base function (in this case, \( x^3 \)) before any transformations are applied.

Explore how to determine the range of the cubic function f(x) = x^3 + 2. This problem highlights the key concepts in analyzing cubic functions and their range, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation