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.