Exploring Functions and Their Inverses

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 is the composition of functions?

Back

The composition of functions, denoted as (f ∘ g)(x), is the application of one function to the results of another. It is defined as f(g(x)).

Front

Find the composition (f ∘ g)(x) if f(x) = x^2 and g(x) = x + 1.

Back

(x + 1)^2.

Front

What is the range of the function f(x) = |x|?

Back

[0, ∞).

Front

What is the inverse of a function?

Back

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

Front

How do you find the inverse of a quadratic function?

Back

To find the inverse of a quadratic function, complete the square and solve for x in terms of y.

Front

Find the inverse of the function f(x) = x^2 + 8x + 16 using completing the square.

Back

f^{-1}(x) = √x - 4.

Front

Find the inverse of the function f(x) = x^2 - 4x + 4 using completing the square.

Back

f^{-1}(x) = √x - 2.

Front

What is the range of the function f(x) = √x?

Back

[0, ∞).

Front

What is a one-to-one function?

Back

A one-to-one function is a function where each output is associated with exactly one input, meaning f(a) = f(b) implies a = b.