Composition and Inverse of Functions

Front

What is the composition of functions?

Back

The composition of functions is the process of applying one function to the results of another function. If f and g are two functions, the composition is denoted as (f ∘ g)(x) = f(g(x)).

Front

What is an inverse function?

Back

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y and produces the original input x.

Front

How do 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 g and f, respectively.

Front

What is the notation for the composition of functions?

Back

The notation for the composition of functions is f(g(x)), which means applying function g to x first, and then applying function f to the result.

Front

What is the first step in finding the inverse of a function?

Back

The first step in finding the inverse of a function is to replace f(x) with y, then solve for x in terms of y.

Front

What does it mean if a function is not one-to-one?

Back

A function is not one-to-one if there are at least two different inputs that produce the same output, which means it does not have an inverse.

Front

What is the graphical representation of a function and its inverse?

Back

The graph of a function and its inverse are symmetric with respect to the line y = x.

Front

How can you verify if a function's inverse is correct?

Back

To verify if a function's inverse is correct, check if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domain.

Front

What is the significance of the domain and range in inverse functions?

Back

The domain of the original function becomes the range of its inverse, and the range of the original function becomes the domain of its inverse.

Front

What is the process to find the inverse of a linear function?

Back

To find the inverse of a linear function, swap x and y, then solve for y to express it in the form of y = mx + b.