Ultimate guide to sqrt(x), sqrt(x+1), sqrt(x-1) with graphs and detailed information

Unlock the Mysteries of the Square Root Graph: โˆšx

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Decoding the Domain: Why x โ‰ฅ 0

  1. Identify the Radicand: In the function โˆšx, the radicand (the number under the square root) is x.
  2. Set the Radicand โ‰ฅ 0: The square root of a negative number is not defined in the real number system, so x must be greater than or equal to zero.
  3. Domain: The domain of โˆšx is x โ‰ฅ 0.

Unlocking the Mysteries: Additional Graph Features

  • Range: The range of โˆšx is y โ‰ฅ 0. The graph starts at the origin (0,0) and extends infinitely upwards and to the right.
  • Quadrants: The graph is located in the 1st quadrant.
  • Curve Shape: The graph is a curve that starts at the origin and extends infinitely to the right, getting gradually flatter as x increases.
  • End Behavior: As x approaches infinity, y also approaches infinity, but at a decreasing rate.


Unlock the Secrets of the Transformed Square Root Graph: โˆš(x-1)

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Deciphering the Domain: Why x โ‰ฅ 1

  1. Identify the Radicand: In the function โˆš(x-1), the radicand (the number under the square root) is x-1.
  2. Set the Radicand โ‰ฅ 0: The square root of a negative number is not defined in the real number system, so x-1 must be greater than or equal to zero.
  3. Solve for x: Solving the inequality x-1 โ‰ฅ 0 gives x โ‰ฅ 1.
  4. Domain: The domain of โˆš(x-1) is x โ‰ฅ 1.

Unlocking the Mysteries: Additional Graph Features

  • Range: The range of โˆš(x-1) is y โ‰ฅ 0. The graph starts at the point (1,0) and extends infinitely upwards and to the right.
  • Quadrants: The graph is located in the 1st quadrant.
  • Curve Shape: The graph is a curve that starts at the point (1,0) and extends infinitely to the right, getting gradually flatter as x increases.
  • End Behavior: As x approaches infinity, y also approaches infinity, but at a decreasing rate.
  • Shift: Compared to the graph of โˆšx, this graph is shifted one unit to the right due to the โ€œ-1โ€ in the radicand.

Unlock the Mysteries of the Transformed Square Root Graph: โˆš(x+1)

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Deciphering the Domain: Why x โ‰ฅ -1

  1. Identify the Radicand: In the function โˆš(x+1), the radicand (the number under the square root) is x+1.
  2. Set the Radicand โ‰ฅ 0: The square root of a negative number is not defined in the real number system, so x+1 must be greater than or equal to zero.
  3. Solve for x: Solving the inequality x+1 โ‰ฅ 0 gives x โ‰ฅ -1.
  4. Domain: The domain of โˆš(x+1) is x โ‰ฅ -1.

Unlocking the Mysteries: Additional Graph Features

  • Range: The range of โˆš(x+1) is y โ‰ฅ 0. The graph starts at the point (-1,0) and extends infinitely upwards and to the right.
  • Quadrants: The graph is located in the 1st and 2nd quadrants.
  • Curve Shape: The graph is a curve that starts at the point (-1,0) and extends infinitely to the right, getting gradually flatter as x increases.
  • End Behavior: As x approaches infinity, y also approaches infinity, but at a decreasing rate.
  • Shift: Compared to the graph of โˆšx, this graph is shifted one unit to the left due to the โ€œ+1โ€ in the radicand.

Unveil the Mysteries of the Transformed Square Root Graph: โˆš(x+1) โ€“ 1

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Deciphering the Domain: Why x โ‰ฅ -1

  1. Identify the Radicand: In the function โˆš(x+1) โ€“ 1, the radicand (the number under the square root) is x+1.
  2. Set the Radicand โ‰ฅ 0: The square root of a negative number is not defined in the real number system, so x+1 must be greater than or equal to zero.
  3. Solve for x: Solving the inequality x+1 โ‰ฅ 0 gives x โ‰ฅ -1.
  4. Domain: The domain of โˆš(x+1) โ€“ 1 is x โ‰ฅ -1.

Unlocking the Mysteries: Additional Graph Features

  • Range: The range of โˆš(x+1) โ€“ 1 is y โ‰ฅ -1. The graph starts at the point (-1,-1) and extends infinitely upwards and to the right.
  • Curve Shape: The graph is a curve that starts at the point (-1,-1) and extends infinitely to the right, getting gradually flatter as x increases.
  • End Behavior: As x approaches infinity, y also approaches infinity.
  • Shift: Compared to the graph of โˆšx, this graph is shifted one unit to the left and one unit down due to the โ€œ+1โ€ in the radicand and the โ€œ-1โ€ subtracted from the function.


Unveil the Mysteries of the Transformed Square Root Graph: โˆš(x-1) + 1

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Deciphering the Domain: Why x โ‰ฅ 1

  1. Identify the Radicand: In the function โˆš(x-1) + 1, the radicand (the number under the square root) is x-1.
  2. Set the Radicand โ‰ฅ 0: The square root of a negative number is not defined in the real number system, so x-1 must be greater than or equal to zero.
  3. Solve for x: Solving the inequality x-1 โ‰ฅ 0 gives x โ‰ฅ 1.
  4. Domain: The domain of โˆš(x-1) + 1 is x โ‰ฅ 1.

Unlocking the Mysteries: Additional Graph Features

  • Range: The range of โˆš(x-1) + 1 is y โ‰ฅ 1. The graph starts at the point (1,1) and extends infinitely upwards and to the right.
  • Curve Shape: The graph is a curve that starts at the point (1,1) and extends infinitely to the right, getting gradually flatter as x increases.
  • End Behavior: As x approaches infinity, y also approaches 1, but at a decreasing rate.
  • Shift: Compared to the graph of โˆšx, this graph is shifted one unit to the right and one unit up due to the โ€œ+1โ€ added to the function and the โ€œ-1โ€ in the radicand.


Uncover the Secrets of the Transformed Square Root Graph: โˆš(x-1) โ€“ 2

Soviet Joke: In Soviet Russia, square root shifts you! ๐Ÿ‡ท๐Ÿ‡บ

Deciphering the Domain: Why x โ‰ฅ 1

  1. Identify the Radicand: In the function โˆš(x-1) โ€“ 2, the radicand (the number under the square root) is x-1.
  2. Set the Radicand โ‰ฅ 0: The square root of a negative number is not defined in the real number system, so x-1 must be greater than or equal to zero.
  3. Solve for x: Solving the inequality x-1 โ‰ฅ 0 gives x โ‰ฅ 1.
  4. Domain: The domain of โˆš(x-1) โ€“ 2 is x โ‰ฅ 1.

Unlocking the Secrets: Additional Graph Features

  • Range: The range of โˆš(x-1) โ€“ 2 is y โ‰ฅ -2. The graph starts at the point (1,-2) and extends infinitely upwards and to the right.
  • Curve Shape: The graph is a curve that starts at the point (1,-2) and extends infinitely to the right, getting gradually flatter as x increases.
  • End Behavior: As x approaches infinity, y also approaches -2, but at a decreasing rate.
  • Shift: Compared to the graph of โˆšx, this graph is shifted one unit to the right and two units down due to the โ€œ+2โ€ added to the function and the โ€œ-1โ€ in the radicand.