MCR3U Grade 11 Functions Math Transformations, Rational Numbers, and Inverse Functions Test

Math Study Notes

Multiplying Radicals

-Numerator multiplies numerator

-Denominator multiplies denominator

-we cross multiply only when separated by an equal sign

Dividing Radicals

-Multiply reciprocal of second term

-Divide out common factors

-State restrictions

Steps:

1) Factor out whatever that’s possible

2) Divide out common factors that are multiplying not adding/subtracting

3) Perform multiplication/division if necessary

4) Simplify answer

Adding/Subtracting

Steps

1) Write the denominator: multiply the missing component from each rationale’s denominator to determine the common denominator

2) Multiply each fraction by any terms needed

3) Add/Subtract numerator, denominator remains the same

4) Expand and simplify numerator

Restrictions

-come from terms eliminated

-come from denominator x value. Denominator cannot be zero

Parent Functions

1) f(x) = x

  1. Linear
  2. D={x|xER}
  3. R={y|yER}
  4. (1,1),(2,2),(3,3),(4,4)..

2)   f(x) = x^2

  1. Quadratic
  2. D={x|xER}
  3. R={y|yER, y>=0}
  4. (1,1),(2,4),(3,9),(4,16)..
  5. Step Pattern: 1,3,5,7..

3)   f(x) = √x

  1. Root Function
  2. D={x|xER,x>=0}
  3. R={y|yER, y>=0}
  4. (1,1),(2,4),(3,9),(4,16)..

4)   f(x) = 1/x

  1. Rational function
  2. D={x|xER, x not= 0}
  3. R={y|yER, y not= 0}
  4. Vertical and Horizontal asymptotes never touch zero

5)   f(x) = |x|

  1. absolute function
  2. D={x|xER}
  3. R={y|yER, y >= 0}

Transformation of Functions:

Stretches/Compression:

-Horizontal Stretch

-Stretch factor determined by reciprocal from equation

-Increases x value by the factor reciprocal

-Sometimes required to factor out x

-stretch > 0 à compression

-stretch < 0 à stretch

-inside with X value

-Vertical Stretch

-stretch > 0 à Stretch

-stretch < 0 à Compression

-outside X value

Translation:

-Horizontal Translation

-Inside with X

-Sometimes need to factor out X to find the actual units

-Opposite sign. X<0 goes right, X>0 goes left

-Vertical Translation

-Outside X

-Goes up and down

-Sign dictates movement

Reflection

-Vertical Reflection

-Up and down on X axis, changes sign of Y values

-Horizontal Reflection

-Left and right along y axis, changes value of x

Inverse Functions

-f(x) and g(x)  are inverse functions of one another

-Reverse each other’s functions

-X and Y are flipped

-Range and Domain flipped

-If graphed, it would be reflected over y=x

Steps

-Switch f(x) and x, but write f(x) as f-1(x).

-solve and simplify

Restrictions

-Restrictions for Inverse must follow restrictions for the original equation