The function given is $f(x) = |x - 5| + 4$.

This is an absolute value function, and we can analyze its properties as follows:

1. Vertex:

• The absolute value function $|x - 5|$ has its vertex at $x = 5$, where the expression inside the absolute value equals zero.
• Therefore, the vertex of the function $f(x)$ is at $(5, 4)$ because when $x = 5$, $f(x) = |5 - 5| + 4 = 0 + 4 = 4$.

2. Shape:

• The graph of $f(x) = |x - 5| + 4$ is a "V" shape, with the point at $(5, 4)$ being the lowest point (vertex) of the graph.
• The function increases linearly with a slope of 1 for $x > 5$ and decreases linearly with a slope of -1 for $x < 5$.

3. Domain:

• The domain of the function is all real numbers, $x \in \mathbb{R}$, because there are no restrictions on the values $x$ can take.

4. Range:

• Since the lowest value of $f(x)$ is 4 (which occurs at $x = 5$), the range of the function is $[4, \infty)$.

5. Intervals of Increase and Decrease:

• The function decreases on the interval $(-\infty, 5]$.
• The function increases on the interval $[5, \infty)$.

Would you like more details on any specific part of this function?

Here are 5 related questions to explore further:

1. How would the graph change if the function were $f(x) = |x - 5| - 4$?
2. What are the effects of shifting the absolute value function horizontally or vertically?
3. How would the function change if it were multiplied by a negative number, such as $f(x) = -|x - 5| + 4$?
4. Can you find the points where the function intersects the x-axis?
5. How would you express this function as a piecewise function?

Tip: When analyzing absolute value functions, always identify the vertex first as it helps to understand the entire graph's behavior.