Composition Of Functions

Front

Define composition of functions.

Back

The composition of functions is an operation that takes two functions, say f and g, and produces a new function by applying one function to the result of the other. It is denoted as (f ∘ g)(x) = f(g(x)).

Front

What is the notation for the composition of functions?

Back

The notation for the composition of functions is f ∘ g, which means 'f composed with g'.

Front

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

Back

(f ∘ g)(2) = f(g(2)) = f(2^2) = f(4) = 2(4) + 3 = 11.

Front

What is the domain of the composition of functions?

Back

The domain of (f ∘ g) 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 - 1 and g(x) = 3x, find (g ∘ f)(1).

Back

(g ∘ f)(1) = g(f(1)) = g(1 - 1) = g(0) = 3(0) = 0.

Front

What is the range of the composition of functions?

Back

The range of (f ∘ g) is the set of all values that f(g(x)) can take as x varies over the domain of g.

Front

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

Back

(f ∘ g)(3) = f(g(3)) = f(3 + 1) = f(4) = 4^2 = 16.

Front

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

Back

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

Front

If f(x) = x^2 - 2 and g(x) = x + 9, find f[g(h(3))] where h(x) = -3x.

Back

f[g(h(3))] = f[g(-9)] = f[6] = 6^2 - 2 = 34.

Front

If f(x) = 3x + 5 and g(x) = x - 7, find f(g(8)).

Back

f(g(8)) = f(8 - 7) = f(1) = 3(1) + 5 = 8.