Sketch a function with the following criteria. Not linear, Increasing of the interval (-infty, infty), and Positive on the interval (3,infty)

To sketch a function that satisfies the given criteria, let's break down what is needed:

1. **Not linear**: The function should not be a straight line.
2. **Increasing on the interval \((-∞, ∞)\)**: The function should be rising throughout its entire domain.
3. **Positive on the interval \((3, ∞)\)**: The function should have values greater than zero for all \(x > 3\).

### Example of Such a Function:
A good example is the **logarithmic function shifted to the right**. Consider the function:

\[ f(x) = \ln(x - 2) + 1 \]

This function has the following properties:
- **Not linear**: The logarithm is a non-linear function.
- **Increasing**: The logarithm function is increasing for \(x > 2\).
- **Positive on \( (3, \infty) \)**: For \(x > 3\), \(f(x)\) is positive.

### Sketch:
- The function starts increasing after \(x = 2\) (since \( \ln(x-2)\) is defined for \(x > 2\)).
- It will cross the x-axis at a point greater than 2.
- For \(x > 3\), the function will always be positive and increasing.

Would you like to see more details on this or have any other questions?

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Here are five related questions you might consider:
1. What are the characteristics of an exponential function that is always increasing?
2. How does a shift in a function affect its graph?
3. What are the properties of the natural logarithm function?
4. How would you find the domain and range of a transformed function?
5. How do different logarithmic bases affect the function's growth?

**Tip**: When sketching functions, always check key points like intercepts, asymptotes, and behavior at infinity!

Sketch a function with the following criteria. Not linear, Increasing on the interval (-∞, ∞), and Positive on the interval (3, ∞).

Learn how to sketch a function that is non-linear, increasing across its entire domain, and positive for x > 3. This problem involves logarithmic functions and transformations, ideal for students in Grades 10-12.

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