Composite and Inverse Functions

Front

What is a composite function?

Back

A composite function is formed when one function is applied to the result of another function, denoted as (f∘g)(x) = f(g(x)).

Front

How do you find (f∘g)(0) if f(x) = 3x + 10 and g(x) = x - 2?

Back

First, find g(0): g(0) = 0 - 2 = -2. Then, find f(-2): f(-2) = 3(-2) + 10 = 4. So, (f∘g)(0) = 4.

Front

What is the definition of an inverse function?

Back

An inverse function reverses the effect of the original function. If f(x) takes x to y, then f⁻¹(y) takes y back to x.

Front

Are the functions f(x) = 2x and g(x) = x^2 + 3 inverses of each other?

Back

No, they are not inverses because f(g(x)) does not equal x.

Front

How do you find f(g(x)) if f(x) = 2x and g(x) = x^2 + 3?

Back

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

Front

What is the notation for composite functions?

Back

The notation for composite functions is (f∘g)(x), which means f(g(x)).

Front

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

Back

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

Front

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

Back

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

Front

What is the domain of a composite function?

Back

The domain of a composite function (f∘g) is the set of all x in the domain of g such that g(x) is in the domain of f.

Front

How do you determine if two functions are inverses?

Back

Two functions f and g are inverses if f(g(x)) = x and g(f(x)) = x for all x in their domains.