Answer to Question #125283 in Algebra for EMMANUEL

Question

Two functions f and h are defined on the set of real numbers, R, by f(x) = (4−x^2)^1/2 and h(x) = (3−x)^1/ 2 (a) Find the domains of f and h. (b) Find foh(x).

Answer 1

Part (a)

f(x)=(4-x^2)^(¹/₂)

Find the domain of f.

f(x)=(4-x^2)^(¹/₂)

4-x^2≥0

4≥x^2

-2≤x≤2

Domain of f: [-2, 2].

 

h(x)=(3-x)^(¹/₂)

Find the domain of h.

h(x)=(3-x)^(¹/₂)

3-x≥0

x≤3

domain of h: (-∞, 3].

 

Part (b)

Find foh(x)

foh(x)=f[h(x)]

and,

f(x)=(4-x^2)^(¹/₂), h(x)=(3-x)^(¹/₂)

 

f(h(x))={4-[(3-x)^(¹/₂)]^2}^(¹/₂)

f[(3-x)^(¹/₂)]=(1+x)^(¹/₂)

Therefore,

foh(x)=(x+1)^(¹/₂)