F(X)= I3X-4I   FIND THE DOMAIN AND RANGE AND GRAPH

Given the function \( F(X) = |3X - 4| \), let's analyze it step by step.

### 1. **Domain**:
The domain of a function represents all the possible values that \( X \) can take such that the function is defined.

- In the case of \( F(X) = |3X - 4| \), since the absolute value function is defined for all real numbers, there are no restrictions on \( X \). 

Thus, the **domain** is:
\[
\text{Domain} = (-\infty, \infty)
\]

### 2. **Range**:
The range of a function is the set of possible values of \( F(X) \) (the output values).

- The absolute value function \( |3X - 4| \) always produces non-negative values, since the absolute value ensures that the result is always \( \geq 0 \).

- The minimum value occurs when \( 3X - 4 = 0 \), which happens at \( X = \frac{4}{3} \). At this point, \( F(X) = 0 \).

- As \( X \) moves away from \( \frac{4}{3} \), the values of \( F(X) \) increase without bound.

Thus, the **range** is:
\[
\text{Range} = [0, \infty)
\]

### 3. **Graph**:

To graph \( F(X) = |3X - 4| \), let's break it down:

- The function represents a V-shaped graph, with the vertex at \( X = \frac{4}{3} \), where \( F(X) = 0 \).

- For values of \( X < \frac{4}{3} \), the graph slopes downward, but due to the absolute value, the function's values remain non-negative.

- For values of \( X > \frac{4}{3} \), the graph slopes upward.

I'll describe the graph's key points:
- Vertex at \( \left( \frac{4}{3}, 0 \right) \)
- Symmetrical around \( X = \frac{4}{3} \)
- Sloping up with a steepness determined by the coefficient 3.

Let me know if you'd like me to generate a visual representation of this graph.

### Recap:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [0, \infty) \)

---

Would you like a visual graph or further details on how to plot the graph? Here are 5 related questions to expand your understanding:

1. What happens if we modify the constant in the absolute value function?
2. How do we find the vertex of other absolute value functions?
3. What is the significance of the coefficient inside the absolute value (e.g., the "3" in \( |3X - 4| \))?
4. How do we solve inequalities involving absolute values like \( |3X - 4| \leq 5 \)?
5. How can we shift or stretch the graph of absolute value functions?

**Tip**: When graphing absolute value functions, always identify the point where the expression inside the absolute value equals zero—it gives the vertex of the graph!

F(X)= |3X - 4| FIND THE DOMAIN AND RANGE AND GRAPH

Explore the domain, range, and graph of the absolute value function F(X) = |3X - 4|. This solution explains key algebraic concepts like the absolute value function, its domain of all real numbers, and its non-negative range. Suitable for high school students in Grades 9-12.

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