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Magazine Chapter 1: Problem 39
Graph the functions. $$y=1+\sqrt{x-1}$$
Short Answer
Expert verified
Plot the function starting at (1,1), extending to the right, with a smooth curve upward.
Step by step solution
01
Identify the Function Type
The given function is \( y = 1 + \sqrt{x-1} \). This is a transformation of the basic square root function \( y = \sqrt{x} \). The transformations applied are a horizontal shift to the right by 1 and a vertical shift up by 1.
02
Find the Domain of the Function
The expression under the square root \( x - 1 \) must be non-negative for the function to be real. Therefore, \( x - 1 \geq 0 \), which simplifies to \( x \geq 1 \). Hence, the domain of the function is \([1, \infty)\).
03
Determine the Range of the Function
Since the smallest value \( \sqrt{x-1} \) can take is 0 (when \( x = 1 \)), then \( y = 1 + 0 = 1 \) is the smallest output value. As \( x \to \infty \), \( \sqrt{x-1} \to \infty \), so \( y \to \infty \). Thus, the range of the function is \([1, \infty)\).
04
Calculate Key Points
Calculate a few key points with specific values of \( x \) to help sketch the graph:- For \( x = 1 \), \( y = 1 + \sqrt{1-1} = 1 \).- For \( x = 2 \), \( y = 1 + \sqrt{2-1} = 2 \).- For \( x = 5 \), \( y = 1 + \sqrt{5-1} \approx 3 \).
05
Sketch the Graph
Start plotting the key points calculated: (1,1), (2,2), and (5,3). Connect these points with a smooth curve, starting at \( x = 1 \) and extending to the right indefinitely since \( x \to \infty \). The curve should resemble the basic \( y = \sqrt{x} \) but shifted and stretched accordingly.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Transformations
Function transformations are ways to modify the basic shape of a function's graph. In this exercise, we start with the basic square root function, which is \( y = \sqrt{x} \). To transform this function, we apply a set of changes:
- Horizontal Shift: The function \( y = 1 + \sqrt{x-1} \) involves moving the graph to the right by one unit. This shift is due to the subtraction inside the square root, changing \( \sqrt{x} \) to \( \sqrt{x-1} \).
- Vertical Shift: Adding 1 to the entire function moves the graph upward by one unit. Thus, \( y = \sqrt{x} \) becomes \( y = \sqrt{x-1} + 1 \).
Domain and Range
Understanding domain and range is crucial for graphing functions effectively. The domain of a function consists of all the possible input values (\( x \)) that result in a real number output.
- For the function \( y = 1 + \sqrt{x-1} \), the domain is determined by the expression under the square root. Since square roots must be non-negative, \( x-1 \geq 0 \), which leads to \( x \geq 1 \). Therefore, the domain is \([1, \infty)\).
- Consider the smallest value \( y \) can take. When \( x = 1 \), \( y = 1 + \sqrt{1-1} = 1 \). As \( x \) increases, \( \sqrt{x-1} \) grows larger, and so does \( y = 1 + \sqrt{x-1} \), approaching infinity. Therefore, the range is \([1, \infty)\).
Key Points Calculation
Calculating key points can be a helpful method in plotting a function's graph accurately. These points ensure the graph begins accurately and is aligned with transformations applied.
To calculate key points for \( y = 1 + \sqrt{x-1} \):
To calculate key points for \( y = 1 + \sqrt{x-1} \):
- At \( x = 1 \): \( y = 1 + \sqrt{1-1} = 1 \).
- At \( x = 2 \): \( y = 1 + \sqrt{2-1} = 2 \).
- At \( x = 5 \): \( y = 1 + \sqrt{5-1} \approx 3 \).
