Math 111 - Sept 14, 2011

2.3) Composition of functions.

Takes the output of one function as the input of a second function...

Learn Math Equation: f(( x)) = 2 x - 7 , Learn Math Equation: g(( x)) =(/ 1,/ x)/

Learn Math Equation: f(( g(( 5)))) means take Learn Math Equation: g(( 5)) =(/ 1,/ 5)/ and put it into Learn Math Equation: f(((/ 1,/ 5)/)) =(/ 2,/ 5)/ - 7 = -(/ 3 3,/ 5)/

Learn Math Equation: f(( g(( 5)))) = f(((/ 1,/ 5)/)) = -(/ 3 3,/ 5)/

Learn Math Equation: g(( f(( 3)))) = g(( 2s* 3 - 7)) = g(( - 1)) = -(/ 1,/ 1)/ = - 1

Learn Math Equation: f(( f(( 2)))) = f(( 2s* 2 - 7)) = f(( - 3)) = 2(( - 3)) - 7 = - 1 3

Learn Math Equation: f(( g(( x)))) = 2(((/ 1,/ x)/)) - 7 =(/ 2,/ x)/ - 7

Learn Math Equation: g(( f(( x)))) =(/ 1,/(( 2 x - 7)))/ =(/ 1,/ 2 x - 7)/

Learn Math Equation: f(( f(( x)))) = 2(( 2 x - 7)) - 7 = 4 x - 1 4 - 7 = 4 x - 2 1

More complex example:

Learn Math Equation: h(( x)) = x - x(^ 2)^ , m(( x)) = 2 x(^ 3)^ + 3

Learn Math Equation: h(( m(( 2)))) = h(( 2s* 2(^ 3)^ + 3)) = h(( 1 9)) = 1 9 - 1 9(^ 2)^ = - 3 4 2

Learn Math Equation: h(( m(( x)))) =(( 2 x(^ 3)^ + 3)) -(( 2 x(^ 3)^ + 3))(^ 2)^ = 2 x(^ 3)^ + 3 -(( 4 x(^ 6)^ + 1 2 x(^ 3)^ + 9)) = - 4 x(^ 6)^ - 1 0 x(^ 3)^ - 6

another:

Learn Math Equation: r(( x)) =(/(r,r x - 2)r,/ x(^ 2)^ + x)/ , Learn Math Equation: s(( x)) = 2 x -(/ 7,/ x)/

Learn Math Equation: r(( s(( x)))) =(/(r,r(( 2 x -(/ 7,/ x)/)) - 2)r,/(( 2 x -(/ 7,/ x)/))(^ 2)^ +(( 2 x -(/ 7,/ x)/)))/ =s.

Learn Math Equation: s(( r(( x)))) = 2(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)) -(/ 7,/(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)))/ =s.

Learn Math Equation: r(( r(( x)))) =(/(r,r(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)) - 2)r,/(((/(r,r x - 2)r,/ x(^ 2)^ + x)/))(^ 2)^ +(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)))/ =s.

Learn Math Equation: r(( r(( r(( x)))))) =(/(r,r(((/(r,r(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)) - 2)r,/(((/(r,r x - 2)r,/ x(^ 2)^ + x)/))(^ 2)^ +(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)))/)) - 2)r,/(((/(r,r(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)) - 2)r,/(((/(r,r x - 2)r,/ x(^ 2)^ + x)/))(^ 2)^ +(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)))/))(^ 2)^ +(((/(r,r(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)) - 2)r,/(((/(r,r x - 2)r,/ x(^ 2)^ + x)/))(^ 2)^ +(((/(r,r x - 2)r,/ x(^ 2)^ + x)/)))/)))/ =s.

Backwards...

Learn Math Equation: w(( x)) =(/ 1,/(r,r 2 x - 7)r)/

Learn Math Equation: f(( x)) = 2 x - 7 , Learn Math Equation: g(( x)) =(/ 1,/(r,r x)r)/

Learn Math Equation: g(( f(( x)))) = w(( x))

or

Learn Math Equation: h(( x)) =(r,r 2 x - 7)r , Learn Math Equation: k(( x)) =(/ 1,/ x)/

Learn Math Equation: k(( h(( x)))) = w(( x))

or

Learn Math Equation: m(( x)) = 2 x , Learn Math Equation: n(( x)) =(/ 1,/(r,r x - 7)r)/

Learn Math Equation: n(( m(( x)))) = w(( x))

Learn Math Graph:334 160 -5.844999999999999 1.0 5.844999999999999 -2.799999999999998 1.0 2.799999999999998 g false Freehand 52 -2.3799999 0.875 -2.345 0.90999997 -2.345 0.90999997 -2.345 0.945 -2.345 0.97999996 -2.3799999 1.015 -2.3799999 1.015 -2.3799999 1.05 -2.415 1.0849999 -2.4499998 1.12 -2.4499998 1.12 -2.485 1.155 -2.5199997 1.19 -2.59 1.19 -2.625 1.225 -2.6599998 1.26 -2.73 1.26 -2.7649999 1.26 -2.8349998 1.26 -2.87 1.225 -2.9399998 1.225 -2.9750001 1.19 -3.01 1.19 -3.0449998 1.155 -3.08 1.12 -3.1149998 1.05 -3.1149998 0.97999996 -3.1499999 0.90999997 -3.1499999 0.84000003 -3.1499999 0.77 -3.1499999 0.66499996 -3.1499999 0.595 -3.1499999 0.48999998 -3.1499999 0.35 -3.1149998 0.24499999 -3.1149998 0.14 -3.08 0.035 -3.0449998 -0.07 -3.0449998 -0.175 -3.01 -0.24499999 -3.01 -0.315 -2.9750001 -0.385 -2.9750001 -0.42000002 -2.9399998 -0.48999998 -2.9399998 -0.525 -2.9399998 -0.56 -2.9399998 -0.56 -2.9399998 -0.595 -2.9399998 -0.63 -2.9399998 -0.63 -2.9399998 -0.66499996 -2.9399998 -0.66499996 Freehand 14 -3.29 0.035 -3.2199998 0.07 -3.2199998 0.07 -3.185 0.07 -3.185 0.07 -3.1499999 0.07 -3.1149998 0.07 -3.08 0.07 -3.0449998 0.07 -2.9750001 0.035 -2.9399998 0.035 -2.87 0.035 -2.7999997 0.035 -2.7999997 0.035 Freehand 47 -1.505 0.28 -1.505 0.35 -1.505 0.35 -1.505 0.35 -1.505 0.385 -1.505 0.385 -1.54 0.42000002 -1.54 0.42000002 -1.5749999 0.42000002 -1.5749999 0.42000002 -1.6099999 0.42000002 -1.645 0.42000002 -1.645 0.385 -1.68 0.35 -1.7149999 0.315 -1.7499999 0.28 -1.7499999 0.21000001 -1.785 0.175 -1.785 0.105000004 -1.785 0.07 -1.785 0.035 -1.7499999 0.0 -1.7149999 -0.035 -1.68 -0.07 -1.6099999 -0.07 -1.5749999 -0.07 -1.505 -0.07 -1.4699999 -0.035 -1.3999999 0.0 -1.365 0.035 -1.3299999 0.07 -1.295 0.14 -1.2599999 0.175 -1.2599999 0.21000001 -1.2599999 0.28 -1.2599999 0.315 -1.2599999 0.35 -1.295 0.385 -1.3299999 0.42000002 -1.3299999 0.42000002 -1.365 0.45499998 -1.3999999 0.45499998 -1.435 0.45499998 -1.4699999 0.45499998 -1.4699999 0.45499998 -1.505 0.45499998 -1.505 0.45499998 Freehand 84 0.17499998 0.315 0.105 0.35 0.07 0.35 0.035 0.35 -0.035 0.35 -0.07 0.315 -0.14 0.315 -0.17499998 0.28 -0.21 0.28 -0.245 0.24499999 -0.28 0.21000001 -0.28 0.175 -0.28 0.14 -0.31499997 0.105000004 -0.31499997 0.07 -0.28 0.035 -0.28 0.0 -0.28 -0.035 -0.245 -0.07 -0.21 -0.07 -0.21 -0.105000004 -0.17499998 -0.105000004 -0.14 -0.105000004 -0.105 -0.105000004 -0.07 -0.035 -0.035 -0.035 0.0 0.035 0.035 0.07 0.035 0.105000004 0.035 0.14 0.07 0.21000001 0.07 0.24499999 0.07 0.28 0.07 0.28 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.315 0.07 0.28 0.07 0.28 0.07 0.24499999 0.07 0.21000001 0.07 0.175 0.07 0.14 0.07 0.105000004 0.07 0.035 0.07 0.0 0.105 -0.14 0.105 -0.21000001 0.105 -0.28 0.105 -0.35 0.105 -0.42000002 0.105 -0.48999998 0.105 -0.56 0.105 -0.595 0.07 -0.73499995 0.07 -0.77 0.035 -0.84000003 0.0 -0.875 0.0 -0.90999997 -0.035 -0.945 -0.105 -0.97999996 -0.14 -0.97999996 -0.17499998 -1.015 -0.21 -1.05 -0.28 -1.05 -0.31499997 -1.05 -0.34999996 -1.05 -0.42 -1.015 -0.455 -1.015 -0.49 -0.97999996 -0.49 -0.97999996 -0.525 -0.97999996 -0.525 -0.945 -0.525 -0.945 -0.525 -0.945 -0.525 -0.945 is Learn Math Equation: f(( g(( x))))

find Learn Math Equation: f(( x)) so that Learn Math Equation: h(( x)) =(((/ x(^ 3)^ + 4,/ x(^ 3)^ - 4)/))(^ 4)^ has the property Learn Math Equation: h(( x)) = f(( g(( x)))) and Learn Math Equation: g(( x)) =(/ x(^ 3)^ + 4,/ x(^ 3)^ - 4)/, ans: Learn Math Equation: f(( x)) = x(^ 4)^

54) A blouse that is size x in Japan is size Learn Math Equation: s(( x)) = x - 3 in the US.

A blouse that is size x in the US is size Learn Math Equation: t(( x)) = x + 4 in Australia.....

From Japan to Australia you do Learn Math Equation: t(( s(( x)))) =(( x - 3)) + 4 = x + 1

2.4a) Graphs of basic functions.

You memorize the shapes of these functions:

Learn Math Equation: y = x

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 1.015 1.015 false \x Point 1.0 1.0 Point 2.0 2.0 Point 3.0 3.0 Point -1.0 -1.0 Point -2.0 -2.0

Learn Math Equation: y = x(^ 2)^

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 1.0 1.0 false \x(^\2)^ Point 0.0 0.0 Point 1.0 1.0 Point 2.0 4.0 Point -1.0 1.0 Point -2.0 4.0

Learn Math Equation: y =(r,r x)r

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 0.7 2.3100002 false (r,r\x)r Point 1.5 0.0 Point 1.5 0.0 Point 0.0 0.0 Point 1.0 1.0 Point 4.0 2.0 Point 0.0 0.0 Color blue

Learn Math Equation: y = x(^ 3)^

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 1.015 1.015 false \x(^\3)^ Color blue Point 0.0 0.0 Color red Point 1.0 1.0 Color red Point -1.0 -1.0 Color red

Learn Math Equation: y =(r 3,r x)r

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 4.095 6.965 false (|\x)|(^(/\1,/\3)/)^s*(/(|\x)|,/\x)/ Color red Point 0.0 0.0 Color blue Point 1.0 1.0 Color blue Point -1.0 -1.0 Color blue

Learn Math Equation: y =(/ 1,/ x)/

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 2.345 4.3050003 false (/\1,/\x)/ Color blue Point 1.0 1.0 Color blue Point -1.0 -1.0 Color blue Point 2.0 0.5 Color blue Point 0.5 2.0 Color blue

Learn Math Equation: y =(| x)|

Learn Math Graph:400 400 -7.0 1.0 7.0 -7.0 1.0 7.0 g true Function 2.485 1.785 false (|\x)| Color blue Point 0.0 0.0 Color blue Point 0.5 0.5 Color blue Point 1.0 1.0 Color blue Point 2.0 2.0 Color blue Point 3.0 3.0 Color blue Point 4.0 4.0 Color blue Point -2.0 2.0 Color blue Point -4.0 4.0 Color blue