Inverses & Composition of Functions

Front

What is the definition of an inverse function?

Back

An inverse function reverses the operation 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 find the inverse of a function algebraically?

Back

To find the inverse of a function f(x), replace f(x) with y, swap x and y, and then solve for y. The resulting equation is the inverse function f⁻¹(x).

Front

What is function composition?

Back

Function composition 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 the composition of functions f(x) = 2x and g(x) = x + 3?

Back

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

Front

If f(x) = x² and g(x) = x + 1, what is (g ° f)(2)?

Back

(g ° f)(2) = g(f(2)) = g(2²) = g(4) = 4 + 1 = 5.

Front

What is the relationship between a function and its inverse?

Back

A function and its inverse are reflections of each other across the line y = x. This means that if (a, b) is a point on the function, then (b, a) is a point on its inverse.

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 f and g.

Front

What is the domain of the inverse function?

Back

The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.

Front

What is the output of f(g(0)) if f(x) = 3x + 10 and g(x) = x - 2?

Back

f(g(0)) = f(-2) = 3(-2) + 10 = 4.

Front

If f(x) = ∛(x) and g(x) = 2x - 5, what is C(x) = |∛(2x - 5) + 3| composed of?

Back

C(x) is composed of j(f(g(x))) where j(x) = |x + 3|.