Inverse Functions F25

Front

What is an inverse function?

Back

An inverse function reverses the operation of the original function. If the function f takes an input x and produces an output y, then the inverse function f^(-1) takes y as input and produces x.

Front

Will the inverse of a function always be a function?

Back

No, the inverse of a function will not always be a function. A function must be one-to-one (each output is produced by exactly one input) for its inverse to also be a function.

Front

What does it mean for a function to be one-to-one?

Back

A function is one-to-one if it never assigns the same value to two different domain elements. In other words, for every y in the range, there is exactly one x in the domain.

Front

How can you determine if two functions are inverses using composition?

Back

To determine if f(x) and g(x) are inverses, compute f(g(x)) and g(f(x)). If both equal x for all x in the domain, then they are inverses.

Front

What is the result of f(g(x)) if f and g are inverse functions?

Back

If f and g are inverse functions, then f(g(x)) = x for all x in the domain of g.

Front

What is the result of g(f(x)) if f and g are inverse functions?

Back

If f and g are inverse functions, then g(f(x)) = x for all x in the domain of f.

Front

What is the notation for the inverse of a function g(x)?

Back

The notation for the inverse of g(x) is g^(-1)(x).

Front

What is the relationship between a function and its inverse in terms of their graphs?

Back

The graph of a function and its inverse are reflections of each other across the line y = x.

Front

If f(x) = 2x + 3, what is f^(-1)(x)?

Back

To find the inverse, swap x and y: y = 2x + 3. Solve for x: x = 2y + 3 -> y = (x - 3)/2. Thus, f^(-1)(x) = (x - 3)/2.

Front

What is the domain of the inverse function if the original function has a range of [0, ∞)?

Back

The domain of the inverse function will be [0, ∞) if the original function has a range of [0, ∞).