Composite Functions

Front

What is 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

How do you denote the composition of two functions f and g?

Back

The composition of two functions f and g is denoted as (f ∘ g)(x), which means 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 difference between (f ∘ g)(x) and (g ∘ f)(x)?

Back

(f ∘ g)(x) = f(g(x)) and (g ∘ f)(x) = g(f(x)). They are generally not equal unless f and g are specific functions.

Front

Given f(x) = 3x - 1 and g(x) = x + 4, find (g ∘ f)(2).

Back

(g ∘ f)(2) = g(f(2)) = g(3(2) - 1) = g(5) = 5 + 4 = 9.

Front

What is the formula for finding (g ∘ g)(x) if g(x) = 2x + 1?

Back

(g ∘ g)(x) = g(g(x)) = g(2x + 1) = 2(2x + 1) + 1 = 4x + 3.

Front

If f(x) = x^2 and g(x) = x + 1, what is (f ∘ g)(x)?

Back

(f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1.

Front

What is the value of g(g(-3)) if g(x) = -x + 5?

Back

g(g(-3)) = g(-(-3) + 5) = g(3 + 5) = g(8) = -8 + 5 = -3.

Front

If f(x) = 4x - 7 and g(x) = -2x + 1, find (f ∘ g)(1).

Back

(f ∘ g)(1) = f(g(1)) = f(-2(1) + 1) = f(-1) = 4(-1) - 7 = -4 - 7 = -11.

Front

What is the output of (g ∘ g)(x) if g(x) = 3x + 7?

Back

(g ∘ g)(x) = g(g(x)) = g(3x + 7) = 3(3x + 7) + 7 = 9x + 21 + 7 = 9x + 28.