Let's analyze and understand the function [tex]\( f(x) = \sqrt{x-9} \)[/tex] step-by-step.
### Step 1: Understand the Function
The function [tex]\( f(x) = \sqrt{x-9} \)[/tex] represents a transformation of the basic square root function [tex]\( \sqrt{x} \)[/tex]. Here, the expression inside the square root, [tex]\( x-9 \)[/tex], indicates a horizontal shift of the basic function to the right by 9 units.
### Step 2: Determine the Domain
For the square root function to be defined, the expression inside the square root must be non-negative. Thus, we require:
[tex]\[ x - 9 \geq 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x \geq 9 \][/tex]
So, the domain of [tex]\( f(x) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x \geq 9 \)[/tex].
### Step 3: Evaluate the Function at Key Points
To get a better understanding of the function, let's evaluate [tex]\( f(x) \)[/tex] at some key points within its domain:
1. At [tex]\( x = 9 \)[/tex]:
[tex]\[ f(9) = \sqrt{9 - 9} = \sqrt{0} = 0 \][/tex]
2. At [tex]\( x = 10 \)[/tex]:
[tex]\[ f(10) = \sqrt{10 - 9} = \sqrt{1} = 1 \][/tex]
3. At [tex]\( x = 13 \)[/tex]:
[tex]\[ f(13) = \sqrt{13 - 9} = \sqrt{4} = 2 \][/tex]
These evaluations give us specific points on the graph of the function: (9, 0), (10, 1), and (13, 2).
### Step 4: Analyze the Behavior of the Function
1. As [tex]\( x \to 9 \)[/tex] from the right:
When [tex]\( x \)[/tex] is just slightly greater than 9, [tex]\( f(x) \)[/tex] will be close to 0 but positive. This establishes that as [tex]\( x \to 9^+ \)[/tex], [tex]\( f(x) \to 0^+ \)[/tex].
2. As [tex]\( x \to \infty \)[/tex]:
Since the square root function grows indefinitely as its input increases, [tex]\( f(x) \)[/tex] will also grow indefinitely as [tex]\( x \)[/tex] increases. Mathematically,
[tex]\[ \lim_{x \to \infty} f(x) = \infty \][/tex]
### Step 5: Graph the Function
To complete our understanding, we can sketch the graph based on the points and behaviors analyzed:
- The graph starts at (9, 0).
- It rises progressively as [tex]\( x \)[/tex] increases, forming a curve typical of the square root function but shifted to the right by 9 units.
### Conclusion
The function [tex]\( f(x) = \sqrt{x-9} \)[/tex]:
- Has a domain [tex]\( [9, \infty) \)[/tex].
- Starts from the point (9, 0) and increases as [tex]\( x \)[/tex] increases.
- Approaches infinitely large values as [tex]\( x \)[/tex] approaches infinity.
This detailed analysis provides a comprehensive understanding of the function [tex]\( f(x) = \sqrt{x-9} \)[/tex].
