Find the domain and range and graph the function f(x)=|3X-4|

Let's find the domain and range of the function \( f(x) = |3x - 4| \) and analyze its behavior for graphing.

### Domain:
The function \( f(x) = |3x - 4| \) involves an absolute value, and absolute value functions are defined for all real numbers. Therefore, the domain is:

\[
\text{Domain} = (-\infty, \infty)
\]

### Range:
An absolute value function always produces non-negative values. The expression \( 3x - 4 \) inside the absolute value can take any real number, but the absolute value ensures the output is never negative. The minimum value of \( |3x - 4| \) occurs when \( 3x - 4 = 0 \), which happens when:

\[
3x - 4 = 0 \quad \Rightarrow \quad x = \frac{4}{3}
\]

At this point, \( f\left( \frac{4}{3} \right) = 0 \), so the function takes values from 0 upwards. Therefore, the range is:

\[
\text{Range} = [0, \infty)
\]

### Graphing:
To graph the function, we can break it into two cases, based on the behavior of the absolute value:

- For \( 3x - 4 \geq 0 \) (i.e., \( x \geq \frac{4}{3} \)), the function is \( f(x) = 3x - 4 \).
- For \( 3x - 4 < 0 \) (i.e., \( x < \frac{4}{3} \)), the function is \( f(x) = -(3x - 4) = -3x + 4 \).

Now, let's summarize:

1. For \( x \geq \frac{4}{3} \), the function behaves as a straight line with slope 3 and intercept -4.
2. For \( x < \frac{4}{3} \), the function behaves as a straight line with slope -3 and intercept 4.

The vertex of the V-shaped graph occurs at \( x = \frac{4}{3} \), with \( f\left( \frac{4}{3} \right) = 0 \).

I can also help generate a graph for you if you'd like.

Would you like me to provide additional details or any questions on this? 

Here are some related questions to further explore the topic:
1. How would the graph change if the function was \( f(x) = |x - 2| \) instead?
2. What is the significance of the vertex in absolute value graphs?
3. How would the range change if the function was \( f(x) = |3x - 4| + 5 \)?
4. Can you express the function \( f(x) = |3x - 4| \) as a piecewise function?
5. What transformations occur if the function becomes \( f(x) = -|3x - 4| \)?

**Tip:** Absolute value functions always form a V-shaped graph, with the vertex corresponding to where the expression inside the absolute value equals zero.

Learn how to find the domain, range, and graph of the function f(x) = |3x - 4|. This step-by-step guide covers key concepts in algebra, including the properties of absolute value functions, and how to represent them graphically. Suitable for students in Grades 9-12.

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