Let f(x) = x^2 – 3x + 8 and

g(x) = x – 5

(a) Find the sum (f + g)(x) and simplify the results.

(b) Find the composite function
(f º g)(x) and simplify the results.

(c) Find (f º g)(2)
Show some work.

(d) Find the composite function of (g º f)(x) and simplify the results.

f(x) = x^2 – 3x + 8

g(x) = x – 5

a) (f+g)(x) = x^2 - 3x + 8 + x - 5

= x^2 -2x + 3

testing:
let x=5
f(5) = 25 - 15 + 8 = 18
g(5) = 0

so f(5) + g(5) = 18+0 = 18
(f+g)(5) = 25 - 10 + 3 = 18

b)
(f º g)(x)
= f(g(x))
= f(x-5)
= (x-5)^2 - 3(x-5) + 8
= x^2 - 13x + 48

c) then (f º g)(2)
= 4 - 26 + 48
= 26

or
f(g(2))
= f(-3)
= 9 + 9 + 8
= 26

d) (g º f)(x)
= g(f(x))
= g(x^2-3x+8)
= x^2-3x+8 - 5
= x^2 - 3x + 3

Wow this is confusing. Can you explain how you got each answer please? Thank you!

To find the sum (f + g)(x), we simply add the two functions f(x) and g(x) together.

(a) (f + g)(x) = f(x) + g(x)

= (x^2 - 3x + 8) + (x - 5)

= x^2 - 3x + 8 + x - 5

= x^2 - 2x + 3

To find the composite function (f º g)(x), we substitute g(x) into f(x) for every occurrence of x.

(b) (f º g)(x) = f(g(x))

= f(x - 5)

= (x - 5)^2 - 3(x - 5) + 8

= x^2 - 10x + 25 - 3x + 15 + 8

= x^2 - 13x + 48

To find (f º g)(2), we substitute 2 for x in the composite function we found in part (b).

(c) (f º g)(2) = 2^2 - 13(2) + 48

= 4 - 26 + 48

= 26

To find the composite function (g º f)(x), we substitute f(x) into g(x) for every occurrence of x.

(d) (g º f)(x) = g(f(x))

= g(x^2 - 3x + 8)

= (x^2 - 3x + 8) - 5

= x^2 - 3x + 8 - 5

= x^2 - 3x + 3