6) For each pair of functions, find

Question

6) For each pair of functions, find

f gx, gf x, f f xand ggx

a)
f x x g x x      ; 4 2  

b)
  ;   1

Answer 1

a)

f[g(x)] = f[g(x)] = f[-4] = -(4) = -4

g[f(x)] = g[f(x)] = g[-2] = -2

f[f(x)] = f[f(x)] = f[-2] = -2

g[g(x)] = g[g(x)] = g[-4] = -4

b)

f[g(x)] = f[g(x)] = f[1] = -1

g[f(x)] = g[f(x)] = g[-1] = 1

f[f(x)] = f[f(x)] = f[-1] = -1

g[g(x)] = g[g(x)] = g[1] = 1

Answer 2

a) Let's first find f[g(x)]:

f[g(x)] = f[g(4)] = f[2] = -2

Next, let's find g[f(x)]:

g[f(x)] = g[f(2)] = g[-2] = 4

Now, let's find f[f(x)]:

f[f(x)] = f[f(2)] = f[-2] = -2

Lastly, let's find g[g(x)]:

g[g(x)] = g[g(4)] = g[2] = 4

Therefore, f[g(x)] = -2, g[f(x)] = 4, f[f(x)] = -2, and g[g(x)] = 4.

b) Since the functions are not defined in the question, we cannot find f[g(x)], g[f(x)], f[f(x)], and g[g(x)].