Cubic Root Function

1. Fundamental Concepts

Definition: A cubic root function is a function of the form
$$f(x) = \sqrt[3]{x}$$

where the output is the value that, when cubed, gives the input.
Domain and Range: Both are all real numbers ( $\mathbb{R}$ ), since cubic roots are defined for negative, zero, and positive inputs.
Behavior: The function is continuous and strictly increasing across its entire domain.

2. Key Concepts

Evaluating Cubic Roots:
$$\sqrt[3]{x^3} = x \quad \text{for all real } x$$
Solving Equations: To solve $\sqrt[3]{x} = a$ , cube both sides:
$$x = a^3$$
Graphing: The graph of $y = \sqrt[3]{x}$ passes through the origin $(0,0)$ and is symmetric with respect to the origin (odd symmetry).

3. Examples
Example 1 (Function Evaluation)
Problem: Find $f(27)$ if $f(x) = \sqrt[3]{x}$ .
Solution:
$$f(27) = \sqrt[3]{27} = 3$$
Example 2 

Solve for $x$ if $\sqrt[3]{2x+1}=5$ .
Step-by-step:
1. Cube both sides: $2x+1 = 5^3 = 125.$
2. Solve: $2x = 124 \Rightarrow x = 62.$
Check: $\sqrt[3]{2\cdot62+1}=\sqrt[3]{125}=5.$ ✓

4. Problem-Solving Tips
Isolate the root: Keep the cubic root expression alone before solving.
Check results: Substitute back into the function for accuracy.
Use graphs: The cubic root graph helps visualize positive and negative solutions.