FUNCTIONS – Composite Functions RULES :

FUNCTIONS – Composite Functions RULES :

FUNCTIONS – Composite Functions RULES : Read as “ f ” at “g” of “x”

FUNCTIONS – Composite Functions RULES : Read as “ f ” at “g” of “x” Read as “ g ” at “f” of “x”

FUNCTIONS – Composite Functions RULES : Read as “ f ” at “g” of “x” Read as “ g ” at “f” of “x” Symbol DOES NOT mean multiply !!!

FUNCTIONS – Composite Functions RULES : Read as “ f ” at “g” of “x” Read as “ g ” at “f” of “x” It is the symbol used to show composite…

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( ƒ ◦ g )(3) if ƒ(x) = 2x – 6 and g(x) = x2

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( ƒ ◦ g )(3) if ƒ(x) = 2x – 6 and g(x) = x2 Using the Rule

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( ƒ ◦ g )(3) if ƒ(x) = 2x – 6 and g(x) = x2 Using the Rule We need to find g(3) first

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( ƒ ◦ g )(3) if ƒ(x) = 2x – 6 and g(x) = x2 Using the Rule Now we place 9 into f(x)

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( g ◦ ƒ )(-2) if ƒ(x) = x2 – x – 12 and g(x) = 0.5x + 4 Using the Rule :

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( g ◦ ƒ )(-2) if ƒ(x) = x2 – x – 12 and g(x) = 0.5x + 4 Using the Rule : First find ƒ( -2 )

FUNCTIONS – Composite Functions RULES : Calculating numeric composites : work “ inside out “ EXAMPLE : Find ( g ◦ ƒ )(-2) if ƒ(x) = x2 – x – 12 and g(x) = 0.5x + 4 Using the Rule : Now substitute -6 in g(x)

RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ ◦ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4

RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ ◦ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4 Substitute into “x”

RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( ƒ ◦ g )(x) if ƒ(x) = x – 1 and g(x) = 0.5x + 4 Combined like terms

RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( g ◦ ƒ )(x) if ƒ(x) = x + 3 and g(x) = 3x2 – 2x + 8

RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( g ◦ ƒ )(x) if ƒ(x) = x + 3 and g(x) = 3x2 – 2x + 8 Substituted ( x + 3 ) for all x’s

RULES : Calculating algebraic composites : substitute the rule of the inside function into the outside function wherever there is a variable EXAMPLE : Find ( g ◦ ƒ )(x) if ƒ(x) = x + 3 and g(x) = 3x2 – 2x + 8

Composite Functions : going backwards Suppose we were given a function h(x) that was the result of a composite operation. How could we determine the two functions that were combined to get that function ? We will always use h(x) = ( f ◦ g )(x) or f[g(x)] Generally, look for things inside parentheses or under roots. What you’ll see is “something” raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the “something”.

Composite Functions : going backwards We will always use h(x) = ( f ◦ g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What you’ll see is “something” raised to a power or under a root. That something becomes our g(x), and then f(x) becomes a simple equation with x replacing the “something”. Example : If h(x) = ( x + 2 )2 was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x)

Composite Functions : going backwards We will always use h(x) = ( f ◦ g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What you’ll see is “something” raised to a power or under a root. Thatsomethingbecomes our g(x), and then f(x) becomes a simple equation with x replacing the “something”. Example : If h(x) = ( x + 2 )2 was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x +2 )2 ** you can see, that x + 2 is raised to the 2nd power ** so x + 2 is our something raised to a power

Composite Functions : going backwards We will always use h(x) = ( f ◦ g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What you’ll see is “something” raised to a power or under a root. Thatsomethingbecomes our g(x), and then f(x) becomes a simple equation with x replacing the “something”. Example : If h(x) = ( x + 2 )2 was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x +2 )2 ** you can see, that x + 2 is raised to the 2nd power ** so x + 2 is our something raised to a power Therefore : g ( x ) = x + 2

Composite Functions : going backwards We will always use h(x) = ( f ◦ g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What you’ll see is “something” raised to a power or under a root. Thatsomethingbecomes our g(x), and then f(x) becomes a simple equation with x replacing the “something”. Example : If h(x) = ( x + 2 )2 was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x +2 )2 ** you can see, that x + 2 is raised to the 2nd power ** so x + 2 is our something raised to a power Therefore : g ( x ) = x + 2 Removing the x + 2 from the parentheses and replacing it with just x creates our f(x)

Composite Functions : going backwards We will always use h(x) = ( f ◦ g )(x) or f[g(x)] Generally, look for things inside parentheses of under roots. What you’ll see is “something” raised to a power or under a root. Thatsomethingbecomes our g(x), and then f(x) becomes a simple equation with x replacing the “something”. Example : If h(x) = ( x + 2 )2 was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) h(x) = ( x +2 )2 ** you can see, that x + 2 is raised to the 2nd power ** so x + 2 is our something raised to a power Therefore : g ( x ) = x + 2 AND f(x) = ( x )2 Removing the x + 2 from the parentheses and replacing it with just x creates our f(x)

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x)

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) Can you see the “something” ??

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) Can you see the “something” ?? It’s x – 5

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) Can you see the “something” ?? It’s x – 5 Therefore : g ( x ) = x – 5

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) g ( x ) = x – 5 Now replace the x – 5 inside each parentheses with just “x”

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) g ( x ) = x – 5 Now replace the x – 5 inside each parentheses with just “x” f ( x ) = 4(x)3+ 2(x)

Composite Functions : going backwards Example : If h(x) = 4( x – 5 )3 + 2( x – 5 ) was created by ( f ◦ g )(x) find the functions f(x) and g(x) that created h(x) g ( x ) = x – 5 f ( x ) = 4( x )3 + 2( x )

Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) What is the “something” ??

Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) What is the “something” ?? 3x – 10 is under a root

Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) What is the “something” ?? 3x – 10 is under a root So : g ( a ) = 3x - 10

Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) g ( x ) = 3x – 10 Now replace the 3x – 10 under the root with just “x”

Composite Functions : going backwards Example : If h(x) = find the f(x) and g(x) that created h(x) g ( x ) = 3x – 10 f ( x ) = Now replace the 3x – 10 under the root with just “x”

FUNCTIONS – Composite Functions RULES :