Question

(b) Given f(x) = x
2
, g(x) = 2x and h(x) = x - 2 be functions from R to R. Find
(i) f ◦ g
(ii) f ◦ h
(iii) h ◦ f
(iv) (f ◦ g) ◦ h
(v) f ◦ (g ◦ h)

Answer 1

(i)

$$f\circ g=f(g(x))=f(2x)=(2x)^2=4x^2$$

(ii)

$$f\circ h=f(h(x))=f(x-2)=(x-2)^2=x^2-4x+4$$

(iii)

$$h\circ f=h(f(x))=h(x^2)=x^2-2$$

(iv)

$$(f\circ g)\circ h=(f\circ g)(h(x))=(f\circ g)(x-2)=$$

$$=f(g(x-2))=f(2(x-2))=(2(x-2))^2=$$

$$=4x^2-16x+16$$

(v)

$$f\circ (g\circ h)=f((g\circ h)(x))=f(g(h(x)))=$$

$$=f(g(x-2))=f(2(x-2))=(2(x-2))^2=$$

$$=4x^2-16x+16$$