Pair of Functions Output Calculations

CHAPTER 07 Further Functions Solutions: Practice Questions 7.6

07 Practice Questions 7.6 1. The diagram shows a pair of functions, whereby the output of the first function f, becomes the input of the second function, g. Find the outputs at point B for the following inputs at A: 1 (i) Substitute 1 in as the input, A, in the first function: 3(1) + 4 = 7 Now, substitute the output of 7 in as the input into the second function: (7)2 5 = 44 Therefore, an input of 1 at A results in an output of 44 at B.

07 Practice Questions 7.6 1. The diagram shows a pair of functions, whereby the output of the first function f, becomes the input of the second function, g. Find the outputs at point B for the following inputs at A: 3 (ii) Substitute 3 in as the input, A, in the first function: 3(3) + 4 = 13 Now, substitute the output of 13 in as the input into the second function: (13)2 5 = 164 Therefore, an input of 3 at A results in an output of 164 at B.

07 Practice Questions 7.6 1. The diagram shows a pair of functions, whereby the output of the first function f, becomes the input of the second function, g. Find the outputs at point B for the following inputs at A: 0 (iii) Substitute 0 in as the input, A, in the first function: 3(0) + 4 = 4 Now, substitute the output of 4 in as the input into the second function: (4)2 5 = 11 Therefore, an input of 0 at A results in an output of 11 at B.

07 Practice Questions 7.6 1. The diagram shows a pair of functions, whereby the output of the first function f, becomes the input of the second function, g. Find the outputs at point B for the following inputs at A: 2 (iv) Substitute 2 in as the input, A, in the first function: 3( 2) + 4 = 2 Now, substitute the output of 2 in as the input into the second function: ( 2)2 5 = 1 Therefore, an input of 2 at A results in an output of 1 at B.

07 Practice Questions 7.6 1. The diagram shows a pair of functions, whereby the output of the first function f, becomes the input of the second function, g. Find the outputs at point B for the following inputs at A: 5 (v) Substitute 5 in as the input, A, in the first function: 3( 5) + 4 = 11 Now, substitute the output of 11 in as the input into the second function: ( 11)2 5 = 116 Therefore, an input of 5 at A results in an output of 116 at B.

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: f (2) (i) f (x) = 2x + 1 f (2) = 2(2) + 1 = 5

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: g ( 1) (ii) g(x) = 5 x g( 1) = 5 ( 1) = 5 + 1 = 6

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: f g ( 1) (iii) From part (ii), g ( 1) = 6 f g ( 1) = f (6) = 2(6) + 1 = 12 + 1 = 13

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: (g f )(2) (iv) From part (i), f (2) = 5 g f (2) = g(5) = 5 5 = 0

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: g ( f (x)) (v) g( f (x)) = g(2x + 1) = 5 (2x + 1) = 5 2x 1 = 4 2x

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: ( f g)( 3) (vi) First we must find the value of g( 3): g( 3) = 5 ( 3) = 5 + 3 = 8 ( f g)( 3) = f(8) = 2(8) + 1 = 17

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: (vii) g2(5) g2(5) = g g(5) First we must find the value of g(5): g(5) = 5 5 = 0 g2(5) = g(0) = 5 0 = 5

07 Practice Questions 7.6 2. f (x) = 2x + 1 and g(x) = 5 x are two functions, where x . Find: (viii) f 2(x) f 2(x) = f(x) f(x) = f(2x + 1) = 2(2x + 1) + 1 = 4x + 2 + 1 = 4x + 3

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: p( 3) (i) p(x) = x2 4 p( 3) = ( 3)2 4 = 9 4 = 5

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: q(5) (ii) q(x) = 7x + 1 q(5) = 7(5) + 1 = 35 + 1 = 36

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: q p(0) (iii) First we must find the value of p(0) p(0) = (0)2 4 = 0 4 = 4 q p(0) = q( 4) = 7( 4) + 1 = 28 + 1 = 27

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: (p q)( 1) (iv) First we need to find the value of q( 1) q( 1) = 7( 1) + 1 = 7 + 1 = 6 (p q)( 1) = p( 6) = ( 6)2 4 = 36 4 = 32

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: (q p)( 2) (v) First we need to find the value of p( 2) p( 2) = ( 2)2 4 = 4 4 = 0 (q p)( 2) = q(0) = 7(0) + 1 = 1

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: q2( 2) (vi) q2( 2) = q q( 2) First we need to find the value of q( 2) q( 2) = 7( 2) + 1 = 14 + 1 = 13 q q( 2) = q( 13) = 7( 13) + 1 = 90

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: (vii) q (p (x)) = q(x2 4) = 7(x2 4) + 1 = 7x2 28 + 1 = 7x2 27

07 Practice Questions 7.6 3. p(x) = x2 4 and q(x) = 7x + 1 are two functions, where x . Find: (viii) p2(x) = p p(x) = p(x2 4) = (x2 4)2 4 = (x2 4)(x2 4) 4 = x4 8x2 + 16 4 = x4 8x2 + 12

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: f (1) (i) When x = 1, y = 3 for the f function f (1) = 3

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: g( 3) (ii) When x= 3, y= 1 for the g function g( 3) = 1

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: f (g( 3)) (iii) g( 3) = 1, from part (ii) fg( 3) = f( 1) = 5

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: (g f )(1) (iv) f (1) = 3 from part (i) (g f )(1) = g(3) = 0

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: g f ( 4) (v) First find f ( 4) = 1, from the graph gf ( 4) = g(1) = 3

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: ( f g )(0) (vi) First find g (0) = 0, from the graph f g(0) = f(0) = 2

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: (vii) g[g (1)] First find g (1) = 3, from the graph gg(1) = g( 3) = 1

07 Practice Questions 7.6 4. The diagram shows the graphs of two functions, f(x) and g(x). Use the graph to find the value of the following: (viii) f 2( 2) f 2( 2) = f f( 2) First find f ( 2) = 2, from the graph f f( 2) = f(2) = 1

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: h( 1) (i) When x= 1, y = 1 for the h function h( 1) = 1

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: g(0) (ii) When x = 0, y = 3 for the g function g(0) = 3

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: g(h( 3)) (iii) First find h( 3) = 4, from the graph g(h( 3)) = g(4) = 0

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: (h g)( 1) (iv) First find g( 1) = 1, from the graph (h g)( 1) = h(1) = 5

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: h(h( 4)) (v) First find h( 4) = 1, from the graph h(h( 4)) = h(1) = 5

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: (g h)(2) (vi) First find h(2) = 1, from the graph (g h)(2) = g( 1) = 1

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: (vii) hg( 4) First find g( 4) = 0, from the graph hg( 4) = h(0) = 2

07 Practice Questions 7.6 5. The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: (viii) g2( 2) g2( 2) = (g g)( 2) First find g( 2) = 0, from the graph (g g)( 2) = g(0) = 3

07 Practice Questions 7.6 6. The diagram shows the graphs of two functions, g (x) and f (x). Use the graph to find the value of the following: g(0) (i) g(0) = 5 f ( 1) (ii) f ( 1) = 3 g (f ( 1)) (iii) = g ( 3) not possible to find. Off the scale.

07 Practice Questions 7.6 6. The diagram shows the graphs of two functions, g (x) and f (x). Use the graph to find the value of the following: ( f g )(3) (iv) f (2 5) = 6 8 g (g (3)) (v) g (2 5) = 5 5 ( g f )(4) (vi) g(3 1) = 2

07 Practice Questions 7.6 6. The diagram shows the graphs of two functions, g (x) and f (x). Use the graph to find the value of the following: (vii) f g ( 2) f (1) = 6 4 (viii) f 2(0) f (3 2) = 6

07 Practice Questions 7.6 7. f : x x3 + x2 3x 5 and g : x 4 3x are two functions, where x : f (2) (i) f(x) = x3 + x2 3x 5 f(2) = (2)3 + (2)2 3(2) 5 = 8 + 4 6 5 = 1

07 Practice Questions 7.6 7. f : x x3 + x2 3x 5 and g : x 4 3x are two functions, where x : fg(0) (ii) g(x) = 4 3x g(0) = 4 3(0) = 4 0 = 4 fg(0) = f(4) = (4)3 + (4)2 3(4) 5 = 64 + 16 12 5 = 63

07 Practice Questions 7.6 7. f : x x3 + x2 3x 5 and g : x 4 3x are two functions, where x : gf ( 2) (iii) f( 2) = ( 2)3+ ( 2)2 3( 2) 5 = 8 + 4 + 6 5 = 3 gf ( 2) = g( 3) = 4 3( 3) = 4 + 9 = 13

07 Practice Questions 7.6 7. f : x x3 + x2 3x 5 and g : x 4 3x are two functions, where x : (f g)(2) (iv) g(2) = 4 3(2) = 4 6 = 2 (f g)(2) = f( 2) = ( 2)3+ ( 2)2 3( 2) 5 = 8 + 4 + 6 5 = 3

07 Practice Questions 7.6 8. f (x) = 6 + 2x x3, g(x) = 3x2 + x 4 and h(x) = 5x + 2 are three functions, where x . Find: h(g(2)) (i) g(x) = 3x2 + x 4 h(g(2)) = h(10) g(2) = 3(2)2 + (2) 4 h(x) = 5x + 2 = 12 + 2 4 = 5(10) + 2 = 10 = 52

07 Practice Questions 7.6 8. f (x) = 6 + 2x x3, g(x) = 3x2 + x 4 and h(x) = 5x + 2 are three functions, where x . Find: f h(0) (ii) h(0) = 5(0) + 2 f h(0) = f (2) = 0 + 2 f(x) = 6 + 2x x3 = 2 = 6 + 2(2) (2)3 = 6 + 4 8 = 2

07 Practice Questions 7.6 8. f (x) = 6 + 2x x3, g(x) = 3x2 + x 4 and h(x) = 5x + 2 are three functions, where x . Find: g(f (1)) (iii) f(1) = 6 + 2(1) (1)3 g(f (1)) = g(7) = 6 + 2 1 g(x) = 3x2 + x 4 = 7 = 3(7)2 + (7) 4 = 147 + 7 4 = 150

07 Practice Questions 7.6 8. f (x) = 6 + 2x x3, g(x) = 3x2 + x 4 and h(x) = 5x + 2 are three functions, where x . Find: (f g)(2) (iv) g(x) = 3x2 + x 4 (f g)(2) = f (10) g(2) = 3(2)2 + (2) 4 = 6 + 2(10) (10)3 = 12 + 2 4 = 6 + 20 1000 = 10 = 974

07 Practice Questions 7.6 8. f (x) = 6 + 2x x3, g(x) = 3x2 + x 4 and h(x) = 5x + 2 are three functions, where x . Find: f g( 1) (v) g(x) = 3x2 + x 4 g( 1) = 3( 1)2+ ( 1) 4 f g( 1) = f ( 2) = 3 1 4 = 6 + 2( 2) ( 2)3 = 2 = 6 4 + 8 = 10