Functions, Inverses, and Compositions

Front

What is a function?

Back

A function is a relation that assigns exactly one output for each input. For example, f(x) = 2x + 3 is a function.

Front

What is the Vertical Line Test?

Back

The Vertical Line Test is a method to determine if a graph represents a function. If any vertical line crosses the graph more than once, it is not a function.

Front

What is 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

How do you find the composition of functions?

Back

To find the composition of functions f and g, denoted as (f ∘ g)(x), you substitute g(x) into f.

Front

What is the composition of p(x) = 3x + 4 and q(x) = 2x²?

Back

q(p(x)) = q(3x + 4) = 2(3x + 4)² = 18x² + 48x + 32.

Front

What does it mean for two functions to be 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.

Front

What is the definition of a one-to-one function?

Back

A one-to-one function is a function where each output is produced by exactly one input. No two different inputs can produce the same output.

Front

What is the difference between a function and a relation?

Back

A relation is a set of ordered pairs, while a function is a specific type of relation where each input has exactly one output.

Front

What is the Horizontal Line Test?

Back

The Horizontal Line Test determines if a function is one-to-one. If any horizontal line intersects the graph more than once, it is not one-to-one.

Front

Can a function have multiple outputs for a single input?

Back

No, a function cannot have multiple outputs for a single input. This would violate the definition of a function.