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 function g first and then applying 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

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 result of g(f(x)) if f(x) = x^2 and g(x) = 3x + 2?

Back

g(f(x)) = g(x^2) = 3(x^2) + 2 = 3x^2 + 2.

Front

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

Back

g(f(x)) = g(x^2) = x^2.

Front

How do you evaluate g(f(-1)) if f(x) = 2x and g(x) = 2x^2 - 1?

Back

First, find f(-1): f(-1) = 2(-1) = -2. Then, find g(-2): g(-2) = 2(-2)^2 - 1 = 8 - 1 = 7.

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. The order of operations matters.

Front

Define function operations.

Back

Function operations include addition, subtraction, multiplication, and division of functions, allowing for the combination of functions to create new functions.

Front

If f(x) = x + 1 and g(x) = x - 1, what is (f + g)(x)?

Back

(f + g)(x) = f(x) + g(x) = (x + 1) + (x - 1) = 2x.