03: Various Types of Functions 

Parent Functions

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Linear Function

f(x) = x

Domain: {x E R}

Range: {y E R}

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Quadratic Function

f(x) = x²

Domain: {x E R}

Range: {y E R/0 ≤ y}

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Square Root Function

f(x) = √x

Domain: {x E R/0 ≤ x}

Range: {y E R/0 ≤ y}

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Reciprocal Function

f(x) = 1/x

Domain: {x E R/x ≠ 0}

Range: {y E R/y ≠ 0}

Asymptote: x = 0, y = 0

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Absolute Value Function

f(x) = |x|

Domain: {x E R}

Range: {y E R/0 ≤ y}

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Cubic Function

f(x) = x³

Domain: {x E R}

Range: {y E R}

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Transformations of Parent Functions

Transformed functions: @@f(x) = a(k(x-d)) + c@@

• Vertical Stretch: ==a==
  • By a factor of….
  • If negative, reflection in the x axis
• Horizontal Stretch: ^^k^^
  • Always 1/k (flipped)
  • By a factor of….
  • If negative, reflection in the y axis
• Vertical Translation: ==c==
  • if positive, moves up
  • If negative, moves down
• Horizontal Translation: ^^d^^
  • Always the opposite sign of what it is in the brackets (sign is flipped)
  • If positive in bracket (so negative alone), then it moves left ( <-- )
  • If negative in bracket (so positive alone), then it moves right ( --> )

Mapping

1. Draw the parent functions’ table of values
2. Create mapping notation using:

Mapping Notation: @@((1/k)x + d, ay + c)@@

3. Apply mapping notation to the parent function and graph (following BEDMAS, order of operations)

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Combinations of Transformations

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Domain and Range of Functions from Equations

• Linear:
  • D: {x E R}
  • R: {y ER}
• Cubic:
  • D: {x E R}
  • R: {y ER}
• Quadratic:
  • D: {x E R}
  • R: {y E R/0 ≤ y}
  • c is the restriction (replacing zero)
• Absolute Value:
  • D: {x E R}
  • R: {y E R/0 ≤ y}
  • c is the restriction (replacing zero)
• Reciprocal:
  • D: {x E R/ x ≠ 0}
  • c replaces the restriction
  • R: {y E R/ y ≠ 0}
  • d replaces the restriction
    • The function cannot touch the asymptote thus the asymptote is our restriction
• Square Root:
  • D: {x E R / 0 ≤ x}
  • Make the number under the square root sign as small as it can be, so zero (because it cannot be negative since you can’t have a negative radicand)
  • R: {y E R / 0 ≤ y}
  • Look at what your lowest y could be as a result of the reduction of x for domain

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The Inverse Function

1. Write the function in x-y notation
2. Swap x and y
3. Solve for y

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