Without using paper and pencil, evaluate each expression given the following functions. ƒ(x)=x+1 and g(x)=x^2 (g∘ƒ)(2)

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. The notation (g∘ƒ)(x) means to apply the function ƒ first and then apply the function g to the result of ƒ. This is crucial for evaluating expressions like (g∘ƒ)(2), as it requires understanding how to substitute and evaluate functions sequentially.

Evaluating Functions

Evaluating functions means finding the output of a function for a given input. For example, to evaluate ƒ(2) using the function ƒ(x) = x + 1, you substitute 2 into the function, resulting in ƒ(2) = 3. This step is essential in function composition, as you first need to evaluate the inner function before applying the outer function.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form g(x) = x^2. Understanding how to evaluate quadratic functions is important when working with compositions, as the output of the inner function can become the input for a quadratic function, affecting the final result significantly.