Functions of f and g

Question

1. Find (f º g)(-7) and (f º g)(-7)

f(x)=2x-1; g(x)=x²-5

 

2. Find (f º g)(x) and (f º g)(x) and the domain of each

f(x)=x²+2; g(x)=√8-x

Thanks! (Please note that "8-x" is inside the radical sign

Answer 1

Hi Nick,

 

1.

f(x) = 2x-1, g(x) = x² - 5

(f º g)(- 7)= f(g(-7)),  g(-7) = (-7)² - 5 = 44

               = f(44)

               = 2(44)-1

               =87

(g º f)(- 7) = g(f(x))

                = g(f(-7)),  f(-7) = 2(-7)-1=-15

                = g(-15)

                =(-15)² - 5 

                = 220

2.

f(x)=x²+2 ;  g(x)=√(8-x)

(f º g)(x) = f(g(x))

              = f(√(8-x))

              = (√(8-x))² + 2

              = 8 - x + 2

              = 6 - x

The domain of (f º g)(x) is (-∞, +∞)

(g º f)(x) = g(f(x))

              = g(x²+2)

              =√(8-x² - 2)

              =√(6 - x²)

For the condition 6 - x² =0, x = ± √6.  For the square root argument to be positive,  x < ± √6

The domain of (g º f)(x) is [-6, 6].