Composition of Functions and Function Operations

Front

Define composition of functions.

Back

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

Front

What is the formula for the composition of two functions f(x) and g(x)?

Back

The formula for the composition of two functions is (f ∘ g)(x) = f(g(x)).

Front

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

Back

f(g(x)) = f(3x + 2) = (3x + 2)^2 = 9x^2 + 12x + 4.

Front

What is the result of g(g(-4)) if g(x) = 3x + 2?

Back

g(g(-4)) = g(3(-4) + 2) = g(-10) = 3(-10) + 2 = -28.

Front

If f(x) = x^2 + 9 and g(x) = x - 9, find f(x) - g(x).

Back

f(x) - g(x) = (x^2 + 9) - (x - 9) = x^2 - x + 18.

Front

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

Back

g(f(x)) = g(2x) = 2(2x)^2 - 1 = 8x^2 - 1.

Front

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

Back

f(g(x)) = f(2x^2 - 1) = 2(2x^2 - 1) = 4x^2 - 2.

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 can yield different results.

Front

Define the term 'function operation'.

Back

Function operations refer to the various ways functions can be combined or manipulated, including addition, subtraction, multiplication, division, and composition.

Front

What is the result of (g ∘ g)(x) if g(x) = 3x + 2?

Back

(g ∘ g)(x) = g(g(x)) = g(3x + 2) = 3(3x + 2) + 2 = 9x + 8.