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Chapter 13: Problem 22

Graph each function. Give the domain and range. \(f(x)=\sqrt{x-2}+2\)

Short Answer

Expert verified
Domain: [2, 鈭�). Range: [2, 鈭�).

Step by step solution

01

- Identify the function

The given function is \( f(x)=\sqrt{x-2}+2 \). This is a square root function shifted horizontally and vertically.
02

- Determine the domain

The domain of a square root function is where the expression under the square root is non-negative. In this case, \( x-2 \geq 0 \). Solving this gives \( x \geq 2 \). So, the domain is \([2, \infty)\).
03

- Determine the range

Since \( \sqrt{x-2} \geq 0 \) for any \( x \geq 2 \), and \( 2 \) is added to this, \( f(x) \geq 2 \). Therefore, the range of the function is \([2, \infty)\).
04

- Plot key points

To graph the function, start by plotting key points. Substitute values of \( x \) in the function to find corresponding \( y \)-values. For example: \( f(2)=2 \), \( f(3)=3 \), \( f(6)=4 \).
05

- Draw the graph

Plot the points from the previous step on a coordinate plane. Connect these points with a smooth curve starting from \( (2, 2) \) and moving rightwards. The curve should start at \( (2, 2) \) and increase slowly as \( x \) increases.
06

- Verify the graph

Check the plotted graph to ensure it meets all the criteria discussed. The graph should start at \( x = 2 \) and there should be no part of the graph for \( x < 2 \). Additionally, the minimum value of \( f(x) \) should be \( 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range
Understanding the domain and range of a function is essential in graphing accurately. The domain is all possible input values (x-values) for the function. For the function \( f(x)=\sqrt{x-2}+2 \), we need the expression under the square root to be non-negative, so \( x-2 \geq 0 \). Solving this gives \( x \geq 2 \), meaning the domain is \[ [2, \infty) \].
The range is all possible output values (y-values). Since \( \sqrt{x-2} \) is always non-negative and we add 2, the smallest value of \( f(x) \) is 2, occurring at \( x = 2 \). Thus, the range is \[ [2, \infty) \]. Always determine the domain first, as it helps in identifying the valid points to plot.
Square Root Function
A square root function typically looks like \( f(x) = \sqrt{x} \), but our function has been shifted. For \( f(x) = \sqrt{x-2} + 2 \):
  • The \( -2 \) inside translates the function horizontally to the right by 2 units.
  • The \( +2 \) outside translates the function vertically upward by 2 units.
This makes our square root function start at the point (2,2) instead of (0,0). The growth is slower because square roots increase gradually. When graphing, always note these transformations to accurately reflect the function on the coordinate plane.
Plotting Points
Plotting points helps to visualize the function effectively. Begin by selecting a few x-values from the domain and calculate the corresponding y-values:
  • For \( x = 2 \), \( f(2)=\sqrt{2-2}+2=2 \)
  • For \( x = 3 \), \( f(3)=\sqrt{3-2}+2=3 \)
  • For \( x = 6 \), \( f(6)=\sqrt{6-2}+2=4 \)
Plot these points on the graph (e.g., (2,2), (3,3), and (6,4)).
After plotting, draw a smooth curve passing through the points, moving rightwards without dipping below the starting point (2,2).
Make sure the curve respects the domain boundaries and accurately represents how the function increases.

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