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F(x) = x − 2, g(x) = 4x + 5

(a)
(f ∘ g)(x) =

(b)
(g ∘ f)(x) =

(c)
(f ∘ g)(5) =

(d)
(g ∘ f)(5) =

F(x) = x − 2, g(x) = 4x + 5 (a) (f ∘ g)(x) = (b) (g ∘ f)(x) = (c) (f ∘ g)(5) = (d-example-1
User Luacassus
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1 Answer

4 votes

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

User Yoges Nsamy
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