ib math aa hl paper 1

Question

The functions f and g are defined such that f(x)=(x+3)/4 and g(x)=8x+5.

(a) Show that (g○f)(x)=2x+11 --○ is composite function

(b) Given that (g○f)^-1(a)=4, find the value of a

Answer 1

f(x) = (x+3)/4

g(x) =8x+5

(gof)(x) = g(f(x)) = 8[(x+3)/4] +5 = 2(x+3)+5 =2x+6+5

= 2x +11

to find the inverse (gof)^-1(x) switch x and y

y=2x+11 becomes

x = 2y+11

solve for the new y

2y=x-11 divide by 2

y =(x-11)/2

(gof)^-1(x) = (x-11)/2

if (gof)^-1(a) = 4

then 4=(a-11)/2

8 = a-11

a =8+11 =19

a =19