Operations and Composition of Functions

Front

Define the 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

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

Back

(f ∘ g)(x) = f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x - 2 + 3 = 2x + 1.

Front

What is the result of (g ∘ f)(x) if f(x) = x^2 and g(x) = 3x + 1?

Back

(g ∘ f)(x) = g(f(x)) = g(x^2) = 3(x^2) + 1 = 3x^2 + 1.

Front

If f(x) = 5x + 7, what is f(2)?

Back

f(2) = 5(2) + 7 = 10 + 7 = 17.

Front

What is the difference between f(g(x)) and g(f(x))?

Back

f(g(x)) and g(f(x)) are different compositions of functions. The order of application matters, leading to potentially different results.

Front

Find f(g(5)) if f(x) = 2x + 5 and g(x) = x - 7.

Back

g(5) = 5 - 7 = -2; then f(g(5)) = f(-2) = 2(-2) + 5 = -4 + 5 = 1.

Front

What is the standard form of a polynomial?

Back

The standard form of a polynomial is expressed as a sum of terms in descending order of the degree of each term, such as a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0.

Front

Perform the operation: (3x^2 + 2x) + (4x^2 - 5x).

Back

(3x^2 + 2x) + (4x^2 - 5x) = (3 + 4)x^2 + (2 - 5)x = 7x^2 - 3x.

Front

What is the result of the operation: 6t^4 - 11t^2 + 7 + 8t^4 - t^2 + 15?

Back

6t^4 + 8t^4 - 11t^2 - t^2 + 7 + 15 = 14t^4 - 12t^2 + 22.