Consider the linear functions fx 8ax b gx 2bx 4a Suppose

Consider the linear functions f(x) = 8ax + b; g(x) = 2bx - 4a. Suppose also that we know and (**) f(1) = 7 and f(g(0)) = -17. Find the composition f(g(x)) as a function of x. Now write the equations (**) in terms of the constants a and b. Now solve to find TWO different pairs of solutions (a_1, b_1) and (a_2, b_2). Now use (iii) to write two pairs of functions f_1 (x), g_1 (x) and f_2 (x), g_2 (x). On one pair of axes graph the equations f_2 (x), g_2 (x). Find and label their intersection. On another pair of axes graph the equations f_2 (x), g_2 (x). Find and label their intersection. Finally, write a chart, which displays for all four functions their values for the x-values -2, -1, 0, 1, 2.

Solution

f(x) = 8ax + b

g(x) = 2bx - 4a

f(g(x)) = 8a[2bx - 4a] + b

=> f(g(x)) = 16abx - 32a² + b

we know that f(1) = 7

therefore 7 = 8a + b .......................[1]

and f(g(0)) = - 17

so

- 32a² + b = - 17

or 17 + b = 32a² ..............................[2]

substitute for b from equation [1]

17 + (7 - 8a) = 32a²

=> 32a² + 8a - 24 = 0

dividing the equation by 8, we get:

=> 4a² + a - 3 = 0

=> 4a² + 4a - 3a - 3 = 0

=> 4a[1 + a] - 3[a + 1] = 0

=> (1+a)(4a - 3) = 0

=> a = -1 or a = 3/4

when a = - 1, b = 7 - 8(-1) = 15

and when a = 3/4, b = 7 - 8(3/4) = 1

so, (a₁,b₁) = (-1,15) and (a₂,b₂) = (3/4, 1)