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Chapter 6: Problem 122

Graph \(f(x)=\sqrt{2-x}\) using the techniques of shifting, compressing or stretching, and reflecting. State the domain and the range of \(f\)

Short Answer

Expert verified
Domain: \( x \leq 2 \). Range: \( [0, \infty) \). The graph is a horizontal shift 2 units to the right and a reflection across the y-axis of \( \sqrt{x} \).

Step by step solution

01

Identify the Base Function

The base function we start with is the square root function, which is \( f(x) = \sqrt{x} \). This is a standard function whose graph we should be familiar with.
02

Apply the Horizontal Shift

Rewrite the given function \( f(x) = \sqrt{2 - x} \) as \( f(x) = \sqrt{-(x - 2)} \). This indicates a horizontal shift of 2 units to the right of the base function \( \sqrt{x} \).
03

Reflect Across the y-Axis

Since the function inside the square root is \(-(x - 2)\), this transformation also involves reflecting the square root function across the y-axis after shifting 2 units to the right. The reflection changes the direction of the graph.
04

Determine the Domain

For \( f(x) = \sqrt{2 - x} \) to be defined, the expression inside the square root must be non-negative: \( 2 - x \geq 0 \). Solve this inequality to find \( x \leq 2 \). Therefore, the domain of \( f(x) \) is all \( x \) values such that \( x \leq 2 \).
05

Determine the Range

The range is determined by the possible values of \( f(x) \). Since \( f(x) \) involves a square root, the output values are always non-negative. Thus, the range is \( f(x) \geq 0 \), or \( [0, \infty) \).
06

Summarize the Transformations and Graph

The original function \( \sqrt{x} \) is horizontally shifted 2 units to the right and then reflected across the y-axis. Sketch the final graph based on these transformations.

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Key Concepts

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

Horizontal Shift
In the context of the function \(f(x) = \sqrt{2 - x}\), one primary transformation is the horizontal shift. A horizontal shift occurs when every point on a function's graph moves left or right by a certain number of units.
To detect a horizontal shift, we look at the inside of the function. The function \(f(x)\) can be rewritten as \(\sqrt{-(x - 2)}\), indicating a shift right by 2 units. This shift means that the graph of \(\sqrt{x}\) now starts from \(x = 2\) instead of \(x = 0\).
Here are key points to remember about horizontal shifts:
  • If we shift a function right, we add to \(x\) inside the function (e.g., \( \sqrt{x - h}\) shifts right by \(h\) units).
  • If we shift a function left, we subtract from \(x\) inside the function (e.g., \( \sqrt{x + h}\) shifts left by \(h\) units).
This is critical for visually graphing functions correctly.
Reflection
Reflections are another essential transformation. Reflection flips the graph over a specific line, often the x-axis or y-axis.
In the function \(f(x) = \sqrt{2 - x}\), the negative sign inside the square root indicates a reflection across the y-axis. This changes the direction of the graph. Instead of the usual \

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