1.3.2 Composite Function (Comparison Method) Example 1 - 3

Example 1
Given

f : x \mapsto h x + k, g : x \mapsto {(x + 1)}^{2} + 4

and

f g : x \mapsto 2 {(x + 1)}^{2} + 5.

Find
(a) the value of g²(2),
(b) the value of h and of k.

Example 2
Given

f : x \mapsto 1 - x

and

g : x \mapsto p x^{2} + q

. If the composite function gf is defined by

g f : x \mapsto 3 x^{2} - 6 x + 5

, find
(a) the value of p and of q,
(b) the value of

g^{2} (- 1)

Example 3:

Given f: x → hx + k and f² : x → 4x + 15.
(a) Find the values of h and of k.
(b) Take h > 0, find the values of x for which f (x²) = 7x

Solution:

(a)

Step 1:
Find f² (x)

Given f (x) = hx + k

f² (x) = ff (x) = f (hx + k)

= h (hx + k) + k

= h²x + hk + k

Step 2:
Compare with given f² (x)

f² (x) = 4x + 15

h²x + hk+ k = 4x + 15

h²= 4

h = ± 2

When, h = 2

hk + k = 15

2k + k = 15

k = 5

When, h = –2

hk + k = 15

–2k + k = 15

k = –15

(b)

h > 0, h = 2, k = 5

Given f (x) = hx + k

f (x) = 2x + 5

f (x²) = 7x

2 (x²) + 5 = 7x

2x² – 7x+ 5 = 0

(2x – 5)(x–1) = 0

2x – 5 = 0 or x –1= 0

x = 5/2
or
x = 1

1.3.2 Composite Function (Comparison Method) Example 1 – 3