Math Problem Statement

Draw the graph of this function

Draw the graph of this function

Solution

Here is the graph of the piecewise function f(x)f(x) as requested. The plot includes:

  • For x<0x < 0: f(x)=(x+2)2f(x) = (x + 2)^2, shown in blue.
  • For x=0x = 0: f(x)=1f(x) = -1, indicated by a red dot at (0,1)(0, -1).
  • For x>0x > 0: f(x)=x3+4f(x) = x^3 + 4, shown in green.

Would you like further explanation on any part of this graph?

Here are some related questions for further exploration:

  1. What is the behavior of the function as xx \to -\infty and xx \to \infty?
  2. How would the graph change if f(x)=(x+2)2f(x) = (x + 2)^2 was defined for x0x \leq 0 instead of x<0x < 0?
  3. What is the derivative of this piecewise function?
  4. Can you determine if the function is continuous at x=0x = 0?
  5. What happens to the graph if we replace x3+4x^3 + 4 with x34x^3 - 4 for x>0x > 0?

Tip: When sketching piecewise functions, plot each interval separately to ensure accuracy at transition points.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Graphing Functions

Formulas

f(x) = (x + 2)^2 for x < 0
f(x) = -1 for x = 0
f(x) = x^3 + 4 for x > 0

Theorems

Continuity of Piecewise Functions
Graph Behavior at Transition Points

Suitable Grade Level

Grades 10-12