Math. Functions

Question

Functions f and g are defined by

f : x = k – x for x ∈ R, where k is a constant,

g : x = 9/x + 2

for x ∈ R, x ≠ –2.

(i) Find the values of k for which the equation f(x) = g(x) has two equal

roots and solve the equation f(x) = g(x) in these cases.

from cie as&a level 9701/01/m/j/06 Q11

Answer 1

f(x) = k - x

g(x) = 9/(x+2)

f(x) = g(x) when k - x = 9 / (x+2)

So, (k-x)(x+2) = 9

-x² + (k-2)x + (2k-9) = 0

So, x = [-(k-2) ± √((k-2)² - 4(-1)(2k-9))] / -2

The roots are equal when (k-2)² + 4(2k-9) = 0

So, k² + 4k - 32 = 0

(k + 8)(k - 4) = 0

k = 4 or -8

If k = 4, then 4-x = 9/(x+2)

(4-x)(x+2) = 9

-x²+2x+8 = 9

x²-2x+1 = 0 So, (x-1)² = 0

x = 1

If k = -8, then -8-x = 9/(x+2)

(-8-x)(x+2) = 9

-x²-10x-16 = 9

x²+10x+25 = 0

(x+5)² = 0

x = -5