Composite Functions Review

Front

Define a composite function.

Back

A composite function is a function that is formed by combining two functions, where the output of one function becomes the input of the other. It is denoted as (f ∘ g)(x) = f(g(x)).

Front

What is the notation for composite functions?

Back

The notation for composite functions is (f ∘ g)(x), which means f(g(x)).

Front

If g(t) = 4t - 2 and h(t) = t^2 + t, find g(h(-3)).

Back

g(h(-3)) = g(6) = 4(6) - 2 = 22.

Front

If f(x) = x + 1 and g(x) = 3x + 5, find f(g(x)).

Back

f(g(x)) = f(3x + 5) = (3x + 5) + 1 = 3x + 6.

Front

If f(x) = x - 2 and g(x) = 2x - 4, find (f ∘ g)(-5).

Back

(f ∘ g)(-5) = f(g(-5)) = f(-14) = -14 - 2 = -16.

Front

If f(t) = -t + 5 and g(t) = 3t + 3, find (f ∘ g)(t).

Back

(f ∘ g)(t) = f(g(t)) = f(3t + 3) = - (3t + 3) + 5 = -3t + 2.

Front

If h(n) = 3n + 1 and g(n) = -2n^2 + 2 + 2n, find (h ∘ g)(n).

Back

(h ∘ g)(n) = h(g(n)) = h(-2n^2 + 2 + 2n) = 3(-2n^2 + 2 + 2n) + 1 = -6n^2 + 6n + 7.

Front

What is the domain of a composite function?

Back

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

Front

Explain how to find the range of a composite function.

Back

To find the range of a composite function (f ∘ g), determine the range of g first, then find the range of f applied to that range.

Front

What is the importance of understanding composite functions in calculus?

Back

Understanding composite functions is crucial in calculus as they are used in the chain rule for differentiation and in integration techniques.