We name Functions f (x) or g(x) etc.

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Section 2 1B: Function Notation Bot of te equations y 2x +1 and y 3x 2 are functions. It is common to ave two or more functions in terms of x in te same problem. If I ask you wat is te value for y if x 5 you would need to ask wic of te two different equations for y do you mean?. It could be confusing to refer to bot of te equations above as y because bot equations are stated in terms of y. We use function notation to avoid tis confusion. We name Functions f (x) or g(x) etc. We name a given function using a letter. Te variable for te domain of te function is ten listed in parenteses after te letter. We will normally use x as te domain variable. Te expression for te range variable is f (x) or g(x) instead of te variable y. Te set of parenteses Does Not Mean to Multiply f (x) is read f of x tis means te name of te function is f and te variable for te domain of te function is x. and te notation for te range variable y is f (x) Wat does f (2) mean? f (2) means tere is a function named f and te value for x is 2 Te value for f (2) can be found by substituting x 2 into te expression for x and evaluating te expression. Example 1 If f (x) 5x + 4 ten f (2) means find te value of f(x) wen x 2 If f (x) 5x + 4 ten f (2) 5(2) + 4 f (2) 14 Mat 370 Section 2 1B Page 1 2018 Eitel

Example 2 (x) is read of x tis means te name of te function is and te variable for te domain of te function is x. and te notation for te range variable y is (x) Wat does (7) mean? (7) means tere is a function named and te value for x is 7 Te value for (7) can be found by substituting x 7 into te expression for x and evaluating te expression If (x) 5x + 8 ten (7) means find te value of (x) wen x 7 If (x) 5x + 8 ten (7) 5(7) + 8 (7) 27 Example 3 Example 4 If f (x) 2x + 3 ten f (4) means If (x) 3x 1 find te value of f(x) ten (5) means wen x 4 find te value of (x) wen x 5 If f (x) 2x + 3 ten If (x) 3x 1 ten f (4) 2(4) + 3 (5) 3(5) 1 11 14 if x 4 ten f(x) or y 11 if x 5 ten (x) or y 14 Mat 370 Section 2 1B Page 2 2018 Eitel

Example 5 Example 6 If f (x) 4x 2 2x If k(x) 6x +1 ten f (3) means ten k(8) means find te value of f(x) find te value of k(x) wen x 3 wen x 8 If f (x) 4x 2 2x ten f (3) 4(3) 2 2(3) 4(9) 6 36 6 If k(x) 6x +1 ten k(8) 6(8) + 1 48 +1 49 f (3) 30 k(8) 7 if x 3 ten f(x) or y 30 if x 8 ten k(x) or y 7 Example 7 Example 8 If k(x) 3x 2 ten k(9) means find te value of k(x) If m(x) x 2 2x wen x 9 ten m(4) means find te value of m(x) If k(x) 3x 2 ten wen x 4 k(9) 3(9) 2 27 2 25 k(9) 5 If m(x) x 2 2x ten m(4) (4) 2 2(4) 16 8 m(4) 8 if x 9 ten k(x) or y 5 if x 4 ten m(x) or y 8 Mat 370 Section 2 1B Page 3 2018 Eitel

Nested Function Notation f ( g(x) ) is read f of g of x te function g(x) is nested inside te function f (x) f ( g(x) ) means tere is a function g(x) tat as an input of x and a output of g(x) and ten te value of g(x) is ten put into f (x) in place of x in f (x) Example 9 If f (x) 2x +1 and g(x) 7x 9 ten find f g(2) f ( g(2) ) means find te value of g(2) by putting 2 into g(x) in place of x and ten putting te value of g(2) If g(x) 7x 9 ten g(2) 7(2) 9 g(2) 14 9 g(2) 5 if f (x) 2x +1 ten put te value of g(2) wic is 5 we now ave te expression f (5) f g(2) f (5) 2(5) +1 f (5) 10 +1 f (5) 11 so f ( g(2) ) 11 Mat 370 Section 2 1B Page 4 2018 Eitel

Example 10 Example 11 If f (x) 3x + 2 and g(x) x + 4 ten find f g( 5) f ( g( 5) ) means find te value of g( 5) by putting 5 into g(x) in place of x and ten putting te value ofg ( 5) If g(x) x + 4 ten g( 5) ( 5) + 4 5 + 4 g( 5) 9 If f (x) 3x + 2 ten put te value of g(2) wic is 9 we now ave te expression f (9) f g(2) f (9) 3(9) + 2 27 + 2 f (17) 25 If g(x) 4x 11 and m(x) 2x 1 ten find g m(3) g( m(3) ) means find te value of m(3) by putting 3 into m(x) in place of x and ten putting te value of m(3) into g(x) in place of x If m(x) 2x 1 ten m(3) 2(3) 1 m(3) 6 91 m(3) 5 if g(x) 4x 11 ten put te value of m(3) wic is 5 into g(x) in place of x we now ave te expression g(5) g m(3) g(5) 4(5) 11 g(5) 20 11 9 g(5) 3 so f ( g( 5) ) 25 so g( m(3) ) 3 Mat 370 Section 2 1B Page 5 2018 Eitel

Example 12 Example 13 If f (x) 4x +1 ten find f f (2) f ( f (2)) means find te value of f(2) by putting 2 and ten putting te value of f(2) back If f (x) 4x +1 f (2) 4(2) +1 f (2) 8 +1 f (2) 9 if f (x) 4x +1 ten put te value of f(2) wic is 9 back we now ave te expression f (9) f f (2) f (9) 4(9) +1 f (9) 36 +1 f (9) 37 so f ( f (2)) 37 If g(x) x 2 +1 ten find g g(2) g( g(2) ) means find te value of g(2) by putting 2 into g(x) in place of x and ten putting te value of g(2) back If g(x) x 2 +1 g(2) (2) 2 +1 g(2) 4 +1 g(2) 5 if g(x) x 2 +1 ten put te value of g(2) wic is 5 back into g(x) in place of x we now ave te expression g(5) g g(2) g(5) (5) 2 +1 g(5) 25 +1 g(5) 26 so g( g(2) ) 26 Mat 370 Section 2 1B Page 6 2018 Eitel

(x+ 2) (x+ 2) means tere is a function named and (x+ 2) can be found by substituting x+ 2 into te expression in place of x and simplifying te expression Example 14 Example 15 If (x) 3x 2 ten (x + 2) means substitute ( x + 2) into te expression 3x 2 in place of x If f (x) 2x + 6 ten f (3x 1) means substitute (3x 1) into te expression 2x + 6 in place of x If (x) 3x 2 ten (x + 2) 3(x + 2) 2 3x + 6 2 If f (x) 2x + 6 ten f (3x 1) 2(3x 1) + 6 6x + 2 + 6 (x + 2) 3x + 4 f (3x 1) 6x + 8 Example 16 Example 17 If m(x) x 2 +1 ten m(x + 3) means substitute (x + 3) into te expression x 2 +1 in place of x If m(x) x 2 +1 ten m(x + 3) (x + 3) 2 +1 x 2 + 6x + 9 +1 If k(x) x 2 x ten find k x +1 means find te value by putting k x +1 of k x +1 (x +1) into k(x) in place of x If k(x) x 2 x k((x +1) (x +1) 2 (x +1) k(x +1) x 2 + 2x +1 x 1 m(x + 3) x 2 + 6x +10 k(x +1) x 2 + x Mat 370 Section 2 1B Page 7 2018 Eitel

Example 18 Example 19 If w(x) x 2 ten find w 8x 24 means find te value by putting w 8x 24 of w 8x 24 (8x 24) into w(x) in place of x x If w(x) 3x + 2 ten find w x 1 means find te value by putting w x 1 of w x 1 (x 1) into w(x) in place of x If w(x) x 2 w(8x 24) w(8x 24 8x 24 2 8(x 3) 2 w(8x 24) 4(x 3) If w(x) w(x 1) w(x 1) x 1 3(x 1) + 2 x 1 3x 3 + 2 x 1 3x 1 Mat 370 Section 2 1B Page 8 2018 Eitel

Te Difference Quotient Function Find an expression for te function f(x + ) f(x) in reduced form for eac of te following functions: Example 1 f(x) 6x Example 2 f(x) 2x + 5 6(x + ) 6x 6 6x + 6 6x 6 2(x + ) + 5 ( 2x + 5) 2x 2 + 2x 5 2 2 Example 3 f(x) 4x 2 4(x + ) 2 4x 2 4 ( x2 + 2x + 2 ) 4x 2 4x2 + 8x + 4 2 4x 2 8x + 42 4(2x + ) 4(2x + ) 8x + 4 Mat 370 Section 2 1B Page 9 2018 Eitel

Te Difference Quotient Function For eac of te following functions find f (x + ) f (x) were 0 Example 1 f(x) x 2 Example 2 f(x) 4x 2 (x + ) 2 x 2 ( x2 + 2x + 2 ) x 2 x2 + 2x + 2 x 2 2x + 2 (2x + ) 4(x + ) 2 4x 2 4 ( x2 + 2x + 2 ) 4x 2 4x2 + 8x + 4 2 4x 2 8x + 42 4(2x + ) 4(2x + ) (2x + ) if 0 2x 8x + 4 If 0 8x Mat 370 Section 2 1B Page 10 2018 Eitel

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