CHAPTER Further Functions Solutions Practice Questions 7 6

CHAPTER Further Functions Solutions: Practice Questions 7. 6 07

Practice Questions 7. 6 07 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: (i) 1 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.

Practice Questions 7. 6 07 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: (ii) 3 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.

Practice Questions 7. 6 07 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: (iii) 0 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 1. Practice Questions 7. 6 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: (iv) − 2 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 1. Practice Questions 7. 6 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: (v) − 5 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 2. (i) Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: f (2) f (x) = 2 x + 1 f (2) = 2(2) + 1 =5

07 2. (ii) Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: g (− 1) g(x) = 5 – x g(− 1) = 5 − (− 1) =5+1 =6

07 2. (iii) Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: f g (− 1) From part (ii), g (− 1) = 6 f g (− 1) = f (6) = 2(6) + 1 = 12 + 1 = 13

07 2. (iv) Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: (g ◦ f )(2) From part (i), f (2) = 5 g ◦ f (2) = g(5) =5– 5 =0

07 2. (v) Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: g ( f (x)) g( f (x)) = g(2 x + 1) = 5 – 2 x – 1 = 4 – 2 x

07 2. (vi) Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: ( f ◦ g)(− 3) 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 2. Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: (vii) g 2(5) = g ◦ g(5) First we must find the value of g(5): g(5) = 5 – 5 =0 g 2(5) = g(0) =5– 0 =5

07 2. Practice Questions 7. 6 f (x) = 2 x + 1 and g(x) = 5 − x are two functions, where x ∈ ℝ. Find: (viii) f 2(x) = f(x) ◦ f(x) = f(2 x + 1) = 2(2 x + 1) + 1 = 4 x + 2 + 1 = 4 x + 3

07 3. (i) Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: p(− 3) p(x) = x 2 – 4 p(− 3) = (− 3)2 – 4 =9– 4 =5

07 3. (ii) Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: q(5) q(x) = 7 x + 1 q(5) = 7(5) + 1 = 35 + 1 = 36

07 3. (iii) Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: q p(0) First we must find the value of p(0) = (0)2 – 4 =0– 4 =– 4 q p(0) = q(− 4) = 7(− 4) + 1 = – 28 + 1 = – 27

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

07 3. (v) Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: (q ◦ p)(− 2) First we need to find the value of p(− 2) = (− 2)2 – 4 =4– 4 =0 (q ◦ p)(− 2) = q(0) = 7(0) + 1 =1

07 3. (vi) Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: q 2(− 2) = q ◦ q(− 2) First we need to find the value of q(− 2) = 7(− 2) + 1 = – 14 + 1 = − 13 q ◦ q(− 2) = q(− 13) = 7(− 13) + 1 = – 90

07 3. Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: (vii) q (p (x)) = q(x 2 – 4) = 7(x 2 – 4) + 1 = 7 x 2 – 28 + 1 = 7 x 2 – 27

07 3. Practice Questions 7. 6 p(x) = x 2 − 4 and q(x) = 7 x + 1 are two functions, where x ∈ ℝ. Find: (viii) p 2(x) = p ◦ p(x) = p(x 2 – 4) = (x 2 – 4)2 – 4 = (x 2 – 4) – 4 = x 4 – 8 x 2 + 16 – 4 = x 4 – 8 x 2 + 12

07 4. (i) Practice Questions 7. 6 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) When x = 1, y = 3 for the f function f (1) = 3

07 4. (ii) Practice Questions 7. 6 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) When x = − 3, y = − 1 for the g function g(− 3) = – 1

07 4. (iii) Practice Questions 7. 6 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)) g(− 3) = – 1, from part (ii) f g(− 3) = f(− 1) = 5

07 4. (iv) Practice Questions 7. 6 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) f (1) = 3 from part (i) (g ◦ f )(1) = g(3) = 0

07 4. (v) Practice Questions 7. 6 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) First find f (− 4) = 1, from the graph g f (− 4) = g(1) = – 3

07 4. (vi) Practice Questions 7. 6 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) First find g (0) = 0, from the graph f ◦ g(0) = f(0) = 2

07 4. Practice Questions 7. 6 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 g g(1) = g(− 3) = – 1

07 4. Practice Questions 7. 6 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 ◦ f(− 2) First find f (− 2) = 2, from the graph f ◦ f(− 2) = f(2) =1

07 5. (i) Practice Questions 7. 6 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) When x = − 1, y = 1 for the h function h(− 1) = 1

07 5. (ii) Practice Questions 7. 6 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) When x = 0, y = 3 for the g function g(0) = 3

07 5. (iii) Practice Questions 7. 6 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)) First find h(− 3) = 4, from the graph g(h(− 3)) = g(4) = 0

07 5. (iv) Practice Questions 7. 6 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) First find g(− 1) = 1, from the graph (h ◦ g)(− 1) = h(1) = – 5

07 5. (v) Practice Questions 7. 6 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)) First find h(− 4) = 1, from the graph h(h(− 4)) = h(1) = – 5

07 5. (vi) Practice Questions 7. 6 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) First find h(2) = − 1, from the graph (g ◦ h)(2) = g(− 1) = 1

07 5. Practice Questions 7. 6 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 5. Practice Questions 7. 6 The diagram shows the graphs of two functions, g(x) and h(x). Use the graph to find the value of the following: (viii) g 2(− 2) = (g ◦ g)(− 2) First find g(− 2) = 0, from the graph (g ◦ g)(− 2) = g(0) = 3

07 6. (i) Practice Questions 7. 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) = – 5 (ii) f (− 1) = – 3 (iii) g (f (− 1)) = g ( − 3) … not possible to find. Off the scale.

07 6. (iv) Practice Questions 7. 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) f (2· 5) = 6· 8 (v) g (g (3)) g (2· 5) = 5· 5 (vi) ( g ◦ f )(4) g(3· 1) = 2

07 6. Practice Questions 7. 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 7. (i) Practice Questions 7. 6 f : x → x 3 + x 2 − 3 x − 5 and g : x → 4 − 3 x are two functions, where x ∈ ℝ: f (2) f(x) = x 3 + x 2 – 3 x – 5 f(2) = (2)3 + (2)2 – 3(2) – 5 =8+4– 6– 5 =1

07 7. (ii) Practice Questions 7. 6 f : x → x 3 + x 2 − 3 x − 5 and g : x → 4 − 3 x are two functions, where x ∈ ℝ: fg(0) g(x) = 4 – 3 x 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 7. (iii) Practice Questions 7. 6 f : x → x 3 + x 2 − 3 x − 5 and g : x → 4 − 3 x are two functions, where x ∈ ℝ: gf (− 2) 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 7. (iv) Practice Questions 7. 6 f : x → x 3 + x 2 − 3 x − 5 and g : x → 4 − 3 x are two functions, where x ∈ ℝ: (f ◦ g)(2) 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 8. (i) Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: h(g(2)) g(x) = 3 x 2 + x – 4 g(2) = 3(2)2 + (2) – 4 = 12 + 2 – 4 = 10 h(g(2)) = h(10) h(x) = 5 x + 2 = 5(10) + 2 = 52

07 8. (ii) Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: f h(0) = 5(0) + 2 =0+2 =2 f h(0) = f (2) f(x) = 6 + 2 x – x 3 = 6 + 2(2) – (2)3 =6+4– 8 =2

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

07 8. (iv) Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: (f ◦g)(2) g(x) = 3 x 2 + x – 4 g(2) = 3(2)2 + (2) – 4 (f ◦g)(2) = f (10) = 6 + 2(10) – (10)3 = 12 + 2 – 4 = 6 + 20 – 1000 = 10 = – 974

07 8. (v) Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: f g(− 1) g(x) = 3 x 2 + 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

07 8. (vi) Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: (h ◦ f)(− 2) f (− 2) = 6 + 2(− 2) – (− 2)3 (h ◦ f )(− 2) = h(10) =6– 4+8 = 5(10) + 2 = 10 = 52

07 8. Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: (vii) h 2(− 3) = h ◦ h(− 3) = 5(− 3) + 2 h ◦ h(− 3) = h(− 13) = 5(− 13) + 2 = – 15 + 2 = – 65 + 2 = – 13 = – 63

07 8. Practice Questions 7. 6 f (x) = 6 + 2 x − x 3, g(x) = 3 x 2 + x − 4 and h(x) = 5 x + 2 are three functions, where x ∈ ℝ. Find: (viii) (f ◦(g ◦ h))(− 1) f ◦ g ◦ h(− 1) = 5(− 1) + 2 = – 5 + 2 = – 3 f ◦ g ◦ h(− 1) = f ◦ g(− 3) = 3(− 3)2 + (− 3) – 4 f ◦ g ◦ h(− 1) = f ◦ g(− 3) = f(20) f (20) = 6 + 2(20) – (20)3 = 27 – 3 – 4 = 6 + 40 – 8000 = 20 = – 7954

07 9. (i) Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: h (g (1)) g(x) = 2 x h (g (1)) = h(2) g(1) = 21 h(x) = 5 x + 2 =2 = 5(2) + 2 = 12

07 9. (ii) Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: g h (0) h(x) = 5 x + 2 h(0) = 5(0) + 2 g h (0) = g(2) = 2 x =0+2 = 22 =2 =4

07 9. (iii) Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: g ( f (1)) f (x) = 5 + 2 x – x 2 f (1) = 5 + 2(1) – (1)2 g ( f (1)) = g(6) =5+2– 1 = 26 =6 = 64

07 9. (iv) Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: (f ◦ g)(2) g(2) = 22 =4 (f ◦ g)(2) = f(4) = 5 + 2(4) – 42 = 5 + 8 – 16 =– 3

07 9. (v) Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: g 2(3) = g ◦ g(3) = 23 =8 g ◦ g(3) = g(8) = 28 = 256

07 9. (vi) Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: (h ◦ f)(− 3) f(− 3) = 5 + 2(− 3) − (− 3)2 (h ◦ f)(− 3) = h(− 10) =5– 6– 9 = 5(− 10) + 2 = – 10 = – 50 + 2 = – 48

07 9. Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: (vii) g f (− 2) = 5 + 2(− 2) – (− 2)2 g f (− 2) = g(− 3) =5– 4– 4 = 2– 3 = =

07 9. Practice Questions 7. 6 f (x) = 5 + 2 x − x 2, h(x) = 5 x + 2 and g(x) = 2 x are three functions, where x ∈ ℝ. Find: (viii) ( f ◦(h ◦g))(− 1) g(− 1) = 2− 1 = ( f ◦(h ◦g))(− 1) = f ◦ h = f(4· 5) = 5 + 2(4· 5) – (4· 5)2 = 5 + 9 – 20· 25 = – 6· 25

07 Practice Questions 7. 6 10. (i) Find the velocity of the object after: (a) two seconds v(x) = 5 + 3 x v(2) = 5 + 3(2) =5+6 = 11 m/s

07 Practice Questions 7. 6 10. (i) Find the velocity of the object after: (b) four seconds v(x) = 5 + 3 x v(4) = 5 + 3(4) = 5 + 12 = 17 m/s

07 Practice Questions 7. 6 10. (i) Find the velocity of the object after: (c) 12 seconds v(x) = 5 + 3 x v(12) = 5 + 36 = 41 m/s

07 Practice Questions 7. 6 10. (i) Find the velocity of the object after: (d) 20 seconds v(x) = 5 + 3 x v(20) = 5 + 3(20) = 5 + 60 = 65 m/s

07 Practice Questions 7. 6 10. (ii) Hence, find the kinetic energy the object has, at each of the times given in part (i). (a) k(v) = 200 v 2 k(11) = 200(11)2 = 200(121) = 24, 200 J

07 Practice Questions 7. 6 10. (ii) Hence, find the kinetic energy the object has, at each of the times given in part (i). (b) k(v) = 200 v 2 k(17) = 200(17)2 = 57, 800 J

07 Practice Questions 7. 6 10. (ii) Hence, find the kinetic energy the object has, at each of the times given in part (i). (c) k(v) = 200 v 2 k(41) = 200(41)2 = 336, 200 J

07 Practice Questions 7. 6 10. (ii) Hence, find the kinetic energy the object has, at each of the times given in part (i). (d) k(v) = 200 v 2 k(65) = 200(65)2 = 845, 000 J

07 Practice Questions 7. 6 10. (iii) Write an expression, in x, for the amount of kinetic energy the object has after x seconds. v(x) = 5 + 3 x k(v) = 200 v 2 Substitute v(x) = 5 + 3 x into the k(v) function k(v) = 200 v 2 k(5 + 3 x) = 200(5 + 3 x)2 (The amount of kinetic energy of the object after x seconds)

07 11. (i) Practice Questions 7. 6 A shop has reduced the price of all laptops by 30%. Find the sale price of a laptop, if it was originally priced at € 750. Discount of 30% Sale price = 70% of € 750 = € 525

07 11. (ii) Practice Questions 7. 6 A shop has reduced the price of all laptops by 30%. On a given day, the shop is offering a special discount of 10% off all sale prices. Find the discounted price of the laptop. 10% off € 525 = € 52· 5 € 525 – € 52· 50 € 472· 50

07 11. (iii) Practice Questions 7. 6 A shop has reduced the price of all laptops by 30%. What is the overall percentage discount applied to the laptop? Discount = € 750 – € 472· 50 = € 277· 50 Overall percentage discount = 100

07 11. (iv) Practice Questions 7. 6 A shop has reduced the price of all laptops by 30%. If x is the original price of a laptop, write an expression for s(x), the price of the laptop in the 30% off sale. s(x) = 0· 7 x (70% of original price x)

07 11. (v) Practice Questions 7. 6 A shop has reduced the price of all laptops by 30%. Write an expression for d(y), the price of the laptop, after the additional 10% discount is applied, where y is the sale price. d(y) = 0· 9 y (90% of sale price y)

07 11. (vi) Practice Questions 7. 6 A shop has reduced the price of all laptops by 30%. Hence, find the composite function, ds(x), the price of a €x laptop, after both discounts have been applied. Use this composite function to verify your answer to part (iii). ds(x) = d ◦ s(x) = d(0· 7 x) = 0· 9(0· 7 x) = 0· 63 x For x = € 750 ds(x) = 0· 63 x = 0· 63(750) = € 472· 50 which verifies the answer to part (iii)