Question

Given f(x) = 5x2 - 4 and g(x)= 5 - 3x2,find the following expressions.(a) (fog)(4)(b) (gof)(2) (c) (f of)(1)(d) (gog)(0)(a) (fog)(4) =(Simplify your answer.)(b) (gof)(2) =(Simplify your answer.)(c) (fof)(1) = (Simplify your answer.)(d) (gog)0) = (Simplify your answer.)

Answer 1

f(x)= 5x² - 4 and g(x)=5 - 3x²

a) (fog)(x) = f(g(x))

= f(5-3x²)

=5(5-3x²)² - 4

Hence, (fog)(x) = 5(5-3x²) - 4

To find (fog)(4) simply substitute x=4 in the above

(fog)(4) = 5[5-3(4)²]² - 4

= 5[ 5-3(16)]² - 4

=5[5 - 48]² - 4

= 5[-43]² - 4

=9245 - 4

=9241

b) (gof)(2)

First find (g o f )(x)

(gof)(x)= g(f(x))

=g(5x² - 4)

=5 - 3(5x² - 4)²

Hence,

(gof)(x) = 5 - 3(5x² - 4)²

Substitute x=2 in the above to get (gof)(2)

(gof)(x) = 5 - 3[5(2)² - 4]²

= 5 - 3[20 - 4 ]²

= 5 - 3(16)²

=5 -768

=-763

c)(fof)(1)

To find the above, we need to first find f[f(x)]

f[f(x)] = f(5x² - 4)

=5(5x² - 4)² - 4

(fof)(x)=5(5x² - 4)² - 4

Substitute x=1 to get (fof)(1)

(fof)(1)=5[5(1)² - 4]² - 4

=5[5 - 4]² - 4

=5(1)² - 4

= 5 - 4

=1

(d) (gog)(0)

First find (gog)(x)

(gog)(x) = g[g(x)]

=g[5 - 3x² ]

=5- 3 [5-3x²]²

(gog)(x) =5- 3 [5-3x²]²

substitute x=0 in the above to get (gog)(0)

(gog)(0) =5 - 3{5-3(0)²]²

=5 - 3[5-0]²

=5 - 3(5)²

=5 - 75

=-70