Composition of Functions

Front

What is the composition of functions?

Back

The composition of functions is a way to combine two functions where the output of one function becomes the input of another. It is denoted as (f ∘ g)(x) = f(g(x)).

Front

If f(x) = 2x + 3 and g(x) = x^2, what is (f ∘ g)(x)?

Back

(f ∘ g)(x) = f(g(x)) = f(x^2) = 2(x^2) + 3 = 2x^2 + 3.

Front

What is the notation for the composition of functions?

Back

The notation for the composition of functions is (f ∘ g)(x), which means f(g(x)).

Front

If f(x) = x + 1 and g(x) = 3x, what is (g ∘ f)(x)?

Back

(g ∘ f)(x) = g(f(x)) = g(x + 1) = 3(x + 1) = 3x + 3.

Front

What is the domain of the composition of functions?

Back

The domain of (f ∘ g)(x) is the set of all x in the domain of g such that g(x) is in the domain of f.

Front

If f(x) = x^2 and g(x) = x - 1, find (f ∘ g)(2).

Back

(f ∘ g)(2) = f(g(2)) = f(2 - 1) = f(1) = 1^2 = 1.

Front

What is the range of the composition of functions?

Back

The range of (f ∘ g)(x) is the set of all possible outputs of f(g(x)).

Front

If h(x) = 2x and k(x) = x + 5, what is (h ∘ k)(x)?

Back

(h ∘ k)(x) = h(k(x)) = h(x + 5) = 2(x + 5) = 2x + 10.

Front

What is the difference between (f ∘ g)(x) and (g ∘ f)(x)?

Back

(f ∘ g)(x) means applying g first and then f, while (g ∘ f)(x) means applying f first and then g. They are generally not equal.

Front

If f(x) = 3x - 2 and g(x) = x^2, find (f ∘ g)(1).

Back

(f ∘ g)(1) = f(g(1)) = f(1^2) = f(1) = 3(1) - 2 = 1.