Composition & Inverses of Functions

Front

What is function composition?

Back

Function composition is the process of applying one function to the results of another function. If f(x) and g(x) are two functions, then the composition of f and g is denoted as (f ∘ g)(x) = f(g(x)).

Front

How do you find f(g(h(x)))?

Back

To find f(g(h(x))), you first evaluate h(x), then substitute that result into g(x), and finally substitute the result of g(x) into f(x).

Front

What is the inverse of a function?

Back

The inverse of a function f(x) is a function that reverses the effect of f. If f(x) = y, then the inverse function f⁻¹(y) = x.

Front

How do you find the inverse of a linear function?

Back

To find the inverse of a linear function f(x) = ax + b, swap x and y, then solve for y. The inverse is f⁻¹(x) = (x - b)/a.

Front

What is the horizontal line test?

Back

The horizontal line test is a method to determine if a function has an inverse that is also a function. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse that is a function.

Front

What is the composition of f(x) = 2x and g(x) = x + 3?

Back

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

Front

How do you evaluate g(f(-10)) for f(x) = 3x + 10 and g(x) = x - 2?

Back

First, find f(-10): f(-10) = 3(-10) + 10 = -20. Then, find g(-20): g(-20) = -20 - 2 = -22.

Front

What is the inverse of f(x) = -4x - 12?

Back

To find the inverse, swap x and y: x = -4y - 12. Solve for y: y = -1/4x - 3. Thus, f⁻¹(x) = -1/4x - 3.

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. This is important for a function to have an inverse.

Front

How do you determine if the inverse of a function is also a function?

Back

Use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the inverse is not a function.