Composite Functions

Front

What is a composite function?

Back

A composite function is a function that is formed by combining two functions, where the output of one function becomes the input of the other. It is denoted as (f∘g)(x) = f(g(x)).

Front

How do you find g(f(x)) for given functions f(x) and g(x)?

Back

To find g(f(x)), substitute f(x) into g(x). This means you take the output of f(x) and use it as the input for g.

Front

Given f(x) = -3x + 7 and g(x) = 2x² - 8, what is g(f(x))?

Back

g(f(x)) = 18x² - 84x + 90.

Front

What is the notation for composite functions?

Back

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

Front

If g(x) = 3x + 4 and h(x) = 3x - 1, how do you find (g∘h)(x)?

Back

To find (g∘h)(x), substitute h(x) into g(x): (g∘h)(x) = g(h(x)) = g(3x - 1) = 9x + 1.

Front

What is the result of f(g(x)) if f(x) = -3x + 7 and g(x) = 2x² - 8?

Back

f(g(x)) = -6x² + 31.

Front

What is the domain of a composite function?

Back

The domain of a composite function (f∘g)(x) 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 evaluate a composite function at a specific value?

Back

To evaluate a composite function at a specific value, first evaluate the inner function at that value, then use the result as the input for the outer function.

Front

What is the difference between f(g(x)) and g(f(x))?

Back

f(g(x)) applies g first and then f, while g(f(x)) applies f first and then g. They generally yield different results unless f and g are inverses.

Front

What is the importance of understanding composite functions in calculus?

Back

Understanding composite functions is crucial in calculus as they are used in the chain rule for differentiation and in integration.