composite and inverse functions flashcard

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 calculate f(g(x)) for given functions f and g?

Back

To calculate f(g(x)), you first evaluate g(x) for a specific x value, and then substitute that result into the function f.

Front

What is the inverse of a function?

Back

The inverse of a function f, denoted as f^(-1), is a function that reverses the effect of f. If f(x) = y, then f^(-1)(y) = x.

Front

How can you determine if two functions are inverses of each other?

Back

Two functions f and g are inverses if f(g(x)) = x and g(f(x)) = x for all x in the domain of f and g.

Front

What is the significance of the composition of functions in real-world applications?

Back

The composition of functions allows us to model complex relationships where the output of one process feeds into another, such as in physics, economics, and engineering.

Front

Given f(x) = 3x + 10 and g(x) = x - 2, find f(g(5)).

Back

First, calculate g(5) = 5 - 2 = 3. Then, f(g(5)) = f(3) = 3(3) + 10 = 19.

Front

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

Back

f(g(x)) = f(x^2 + 3) = 2(x^2 + 3) = 2x^2 + 6.

Front

If f(x) = x^2 + x and g(x) = 3x + 1, what is f(g(4))?

Back

g(4) = 3(4) + 1 = 13. Then, f(g(4)) = f(13) = 13^2 + 13 = 182.

Front

What does it mean if two functions are not inverses of each other?

Back

If two functions are not inverses, it means that applying one function after the other does not return the original input, i.e., f(g(x)) ≠ x.

Front

How do you find the composition of two functions algebraically?

Back

To find the composition of two functions f and g, substitute g(x) into f. This is done by replacing every instance of x in f with g(x).