(f ∘ g)(x)

Definition

The composition of two functions, $f$ and $g$, evaluated at the input $x$. This represents the result of first applying the function $g$ to the input $x$, and then applying the function $f$ to the result of $g(x)$.

5 Must Know Facts For Your Next Test

1. The notation $(f ∘ g)(x)$ represents the composition of the functions $f$ and $g$, where the output of $g(x)$ becomes the input for $f(x)$.
2. Composing functions allows you to create new functions by combining the operations of existing functions.
3. The domain of the composite function $(f ∘ g)(x)$ is the set of all $x$ values for which $g(x)$ is in the domain of $f$.
4. Composing functions is not commutative, meaning $(f ∘ g)(x) \neq (g ∘ f)(x)$ in general.
5. Inverse functions can be used to 'undo' the effect of a composite function, so that $(f ∘ g)^{-1}(x) = g^{-1}(f^{-1}(x))$.

Review Questions

• Explain the process of evaluating a composite function $(f ∘ g)(x)$.
  • To evaluate a composite function $(f ∘ g)(x)$, you first evaluate the inner function $g(x)$, and then use the result as the input for the outer function $f(x)$. This means that you are applying the function $f$ to the output of $g$. The final result is the value of $f(g(x))$, which represents the composition of the two functions.
• Describe the relationship between the domains of the component functions $f$ and $g$ and the domain of the composite function $(f ∘ g)(x)$.
  • The domain of the composite function $(f ∘ g)(x)$ is the set of all $x$ values for which $g(x)$ is in the domain of $f$. In other words, the composite function $(f ∘ g)(x)$ is defined only for those $x$ values where the output of $g(x)$ is a valid input for the function $f$. This means that the domain of the composite function is the intersection of the domains of the component functions $f$ and $g$.
• Explain why the composition of functions is generally not commutative, and provide an example to illustrate this.
  • The composition of functions is generally not commutative, meaning that $(f ∘ g)(x) \neq (g ∘ f)(x)$ in most cases. This is because the order in which the functions are applied matters. When composing $f$ and $g$, the output of $g(x)$ becomes the input for $f(x)$, whereas when composing $g$ and $f$, the output of $f(x)$ becomes the input for $g(x)$. These two different sequences of operations can lead to different results. For example, if $f(x) = x^2$ and $g(x) = x + 1$, then $(f ∘ g)(x) = (x + 1)^2$, but $(g ∘ f)(x) = \sqrt{x} + 1$, which are not equal.

Related terms

Composite Function:

A new function created by combining two or more existing functions, where the output of one function becomes the input of the next function.

Domain and Range: The set of all possible input values (domain) and output values (range) for a function.

Inverse Function:

A function that 'undoes' the operation of another function, so that when the two functions are applied in succession, the original input is recovered.