Math Problem Statement

The image shows a graph on a coordinate plane and asks to match it with one of the given functions based on its appearance. The options are: 1. \( f(x) = \sqrt{x + 2} + 1 \) 2. \( f(x) = \sqrt{x + 2} - 1 \) 3. \( f(x) = \sqrt{x - 2} + 1 \) 4. \( f(x) = \sqrt{x - 1} + 2 \).

Solution

The image shows a graph on a coordinate plane and asks to match it with one of the given functions based on its appearance. The options are:

  1. f(x)=x+2+1f(x) = \sqrt{x + 2} + 1
  2. f(x)=x+21f(x) = \sqrt{x + 2} - 1
  3. f(x)=x2+1f(x) = \sqrt{x - 2} + 1
  4. f(x)=x1+2f(x) = \sqrt{x - 1} + 2

Let's analyze the graph and functions:

  • The graph starts at x=2x = -2, and increases. The point (2,1)(-2, 1) is highlighted as the start of the curve, which is consistent with a square root function that has been shifted.
  1. Option 1: f(x)=x+2+1f(x) = \sqrt{x + 2} + 1

    • The domain is x2x \geq -2, because the expression inside the square root must be non-negative (x+20x + 2 \geq 0).
    • The function is shifted 2 units to the left and 1 unit up.
    • The starting point would be (2,1)(-2, 1), which matches the given graph.
  2. Option 2: f(x)=x+21f(x) = \sqrt{x + 2} - 1

    • The domain is the same as Option 1 (x2x \geq -2), but the graph would start at (2,1)(-2, -1), which does not match the graph.
  3. Option 3: f(x)=x2+1f(x) = \sqrt{x - 2} + 1

    • The domain is x2x \geq 2, so it does not match the graph which starts at x=2x = -2.
  4. Option 4: f(x)=x1+2f(x) = \sqrt{x - 1} + 2

    • The domain is x1x \geq 1, which does not match the graph either.

Conclusion:

The graph matches Option 1: f(x)=x+2+1f(x) = \sqrt{x + 2} + 1.

Would you like further details on any of the functions, or help with other related questions?

Here are some related questions:

  1. How do we derive the domain of a square root function?
  2. What happens to a graph when the function is shifted horizontally or vertically?
  3. How can we determine the range of a square root function?
  4. Why do square root functions have restricted domains?
  5. How can we find the inverse of a square root function?

Tip: Always check the domain and range of a function when comparing it to a graph, as they are key to identifying the correct match.

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Math Problem Analysis

Mathematical Concepts

Function Analysis
Domain and Range
Square Root Functions
Graph Shifts

Formulas

\( f(x) = \sqrt{x + a} + b \)

Theorems

Domain Restriction of Square Root Functions

Suitable Grade Level

Grades 9-11