The given function is:

\[
f(x) = (x + 4)^3 - 5
\]

To find the range, let's analyze the function step-by-step:

1. The expression \( (x + 4)^3 \) represents a cubic function, which has a domain of all real numbers (\(\mathbb{R}\)) and a range of all real numbers (\(\mathbb{R}\)) because cubic functions increase without bound in both directions.
   
2. Shifting this cubic function by 4 units to the left (\(x + 4\)) does not affect the range, so the range of \( (x + 4)^3 \) is still all real numbers.

3. The next step is the subtraction of 5. Subtracting a constant shifts the entire graph down by 5 units. Thus, the range of \( f(x) \) is also all real numbers (\(\mathbb{R}\)).

Therefore, the range of the function \( f(x) = (x + 4)^3 - 5 \) is:

\[
\text{Range of } f(x) = (-\infty, \infty)
\]

Would you like more details or further explanations?

Here are 5 related questions:

1. What is the domain of a cubic function?
2. How do transformations affect the graph of a cubic function?
3. How would adding or subtracting a constant inside the cube affect the range?
4. Can a cubic function have a limited range?
5. What are the key points to consider when graphing a cubic function?

**Tip:** Cubic functions are always increasing or decreasing, which means they have no maximum or minimum points.

Explore how to determine the range of the cubic function f(x) = (x + 4)^3 - 5 with a clear step-by-step analysis. This guide covers key concepts like cubic function transformations and their impact on the range. Ideal for high school 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