Compositions and Operations with Functions

Front

What is a function?

Back

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.

Front

What does it mean to compose functions?

Back

Composing functions means applying one function to the result of another function, denoted as (f ∘ g)(x) = f(g(x)).

Front

How do you find f(g(x)) if f(x) = 2x and g(x) = x² + 3?

Back

To find f(g(x)), substitute g(x) into f: f(g(x)) = f(x² + 3) = 2(x² + 3) = 2x² + 6.

Front

What is the sum of two functions?

Back

The sum of two functions f and g, denoted (f + g)(x), is defined as (f + g)(x) = f(x) + g(x).

Front

How do you evaluate (f + g)(2) if f(x) = 3x² + 1 and g(x) = 4x + 5?

Back

(f + g)(2) = f(2) + g(2) = (3(2)² + 1) + (4(2) + 5) = 26.

Front

What is the notation for function composition?

Back

Function composition is denoted by (f ∘ g)(x) or f(g(x)).

Front

How do you evaluate g[h(-2)] if h(x) = 3x + 4 and g(x) = 2x - 1?

Back

First, find h(-2): h(-2) = 3(-2) + 4 = -2. Then, find g(-2): g(-2) = 2(-2) - 1 = -5.

Front

What is the output of g(4x) if g(x) = 3x + 4?

Back

g(4x) = 3(4x) + 4 = 12x + 4.

Front

What is the difference of two functions?

Back

The difference of two functions f and g, denoted (f - g)(x), is defined as (f - g)(x) = f(x) - g(x).

Front

How do you find g(h(x)) if g(x) = 3x + 4 and h(x) = 2x - 1?

Back

To find g(h(x)), substitute h(x) into g: g(h(x)) = g(2x - 1) = 3(2x - 1) + 4 = 6x + 1.