Composite Functions

Front

What is a composite function?

Back

A composite function is a function that is formed by combining two functions, denoted as (f∘g)(x) = f(g(x)).

Front

How do you find g(f(x))?

Back

To find g(f(x)), first evaluate f(x) and then substitute that result into 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 composite functions?

Back

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

Front

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

Back

(g∘f)(2) = g(f(2)) = g(2 + 1) = g(3) = 3(3) = 9.

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

If h(x) = x^2 and k(x) = 2x - 1, find h(k(3)).

Back

h(k(3)) = h(2(3) - 1) = h(6 - 1) = h(5) = 5^2 = 25.

Front

What is the range of a composite function?

Back

The range of a composite function (f∘g) is the set of all possible outputs f(g(x)) for x in the domain of g.

Front

If f(x) = x - 4 and g(x) = 5x, find (f∘g)(1).

Back

(f∘g)(1) = f(g(1)) = f(5(1)) = f(5) = 5 - 4 = 1.

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, leading to different results.