Hello!
In this question, we are solving "Composite Functions".
We were given the functions:
- f(x) = x − 2
- g(x) = 4x + 5
We will use these to solve each part of the question.
Solve:
Whenever we see "(f ∘ g)", it means f of g. Another way to write (f ∘ g) is f(g(x)) where g(x) is plugged into the "x" variable of our f(x) function.
(a)
(f ∘ g)(x) = f(g(x))
Plug the g(x) function into the "x" variable of the f(x) function.
(f ∘ g)(x) = 4x + 5 − 2
Simplify.
(f ∘ g)(x) = 4x + 3
(b)
For question "b", it is the same concept but reversed. Plug the f(x) function into the "x" variable of the g(x) function.
(g ∘ f)(x) = g(f(x)) = 4(x − 2) + 5
Simplify.
(g ∘ f)(x) = g(f(x)) = 4(x − 2) + 5
Use the distributive property to distribute the 4.
(g ∘ f)(x) = g(f(x)) = 4x − 8 + 5
(g ∘ f)(x) = g(f(x)) = 4x − 3
(c)
For problem "c", we will plug our number into the "x" variable after compositing the functions. Use the same concept described above.
(f ∘ g)(5) = f(g(x))
(f ∘ g)(5) = 4x + 5 − 2
Simplify.
(f ∘ g)(5) = 4x + 3
Plug in "5" to "x" and simplify.
(f ∘ g)(5) = 4(5) + 3
(f ∘ g)(5) = 23
(d)
Problem "d" is just like problem "c", but the functions are reversed
(g ∘ f)(5) = g(f(x)) = 4(x − 2) + 5
Use the distributive property to distribute the 4.
(g ∘ f)(5) = g(f(x)) = 4x − 8 + 5
(g ∘ f)(5) = g(f(x)) = 4x − 3
Plug in "5" to "x" and simplify.
(g ∘ f)(5) = 4(5) − 3
(g ∘ f)(5) = 17
Answer:
(a) 4x + 3
(b) 4x − 3
(c) 23
(d) 17