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_1 (x), g_1 (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(1) = 7 and f(g(0)) = -17

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

= 16abx - 32a^2 +b

ii) f(1) = 7   ; 7 = 8a +b

f(g(0)) = -17 ; g(0) = -4a

f(g(0)) = 8a(-4a) + b = -17 ; -32a^2 +b = -17

So, -32a^2 + 7 -8a = -17

-32a^2 - 8a + 24 =0

-4a^2 -a +3 =0

a = (1 +/- sqrt( 1+ 48))/-8 = (1 +/- 7)/-8

iii) a1 = -1 ; a2 = 3/4

So, b1 = 7 -8a1 = 7 +8 = 15

b2 = 7 - 8a2 = 7 - 6 = 1

iv) 8ax + b ; g(x) = 2bx - 4a

f1(x) = -8x +15 ; f2(x) = 8(3/4)x + 15 = 6x +15

g1(x) = 30x + 4 ; g2(x) = 2x - 3