Composite & Inverse 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 j(h(x)) if j(x) = ∛(x-1) and h(x) = x³ + 1?

Back

To find j(h(x)), substitute h(x) into j(x): j(h(x)) = j(x³ + 1) = ∛((x³ + 1) - 1) = ∛(x³) = x.

Front

What is the formula for finding the inverse of a function?

Back

To find the inverse of a function f(x), swap x and y in the equation y = f(x) and solve for y. The result is f⁻¹(x).

Front

Given f(x) = x² - 5, what is f⁻¹(x)?

Back

The inverse f⁻¹(x) is √(x + 5) for x ≥ 5, since the original function is not one-to-one without restricting its domain.

Front

What is the significance of the inverse of a function?

Back

The inverse of a function reverses the effect of the original function, meaning if f(a) = b, then f⁻¹(b) = a.

Front

How do you find g(f(x)) if g(x) = -3x and f(x) = x² + 2?

Back

To find g(f(x)), substitute f(x) into g(x): g(f(x)) = g(x² + 2) = -3(x² + 2) = -3x² - 6.

Front

What is the inverse of the coordinates (1, 2), (3, 8), (-3, 6)?

Back

The inverse of the coordinates is (2, 1), (8, 3), (6, -3).

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

What is the range of a function?

Back

The range of a function is the set of all possible output values (y-values) that the function can produce.

Front

How do you determine if a function has an inverse?

Back

A function has an inverse if it is one-to-one, meaning it passes the horizontal line test (no horizontal line intersects the graph more than once).