Function Composition Lesson

We previously learned about operations on function, which allowed us to combine two functions through addition, subtraction, multiplication, and division to create a new function. Here, we will look at another way of combining two functions to create a new function, known as function composition.

Function Composition

Given two functions f and g, the composite function f ∘ g, which is also known as the composition of f and g, is defined by: $$(f ∘ g)(x) = f(g(x))$$ Either side can be read as "f composed with g of x" or "f of g of x".
Note: The symbol "∘" is not a multiplication symbol. It is used for function composition. In general, the composite function f ∘ g is not the same as the product fg.
Example #1: Find f ∘ g. $$f(x) = \sqrt{x}$$ $$g(x) = x^2 + 5$$ To find f ∘ g, we want to plug g(x) in for x in f(x). $$(f ∘ g)(x) = f(g(x)) = \sqrt{x^2 + 5}$$

Domain of the Composite Function f ∘ g

To find the domain for f ∘ g, we state that we have the set of all x such that:

1. x is in the domain of g and
   • If x is not in the domain of g, then it can't be in the domain of f ∘ g
2. g(x) is in the domain of f
   • Any x-value for which g(x) is not in the domain of f can't be in the domain of f ∘ g

f(g(x)) is defined whenever g(x) and f(g(x)) are both defined. In other words, the domain of f(g(x)) will need to exclude any values for x where g(x) is undefined and also where f(g(x)) is undefined.
Example #2: Find the function f ∘ g and the domain. $$f(x) = 2x^2$$ $$g(x) = x + 7$$ To find f ∘ g, we plug g(x) in for x in f(x): $$(f ∘ g)(x) = f(g(x))$$ $$= f(x + 7)$$ $$= 2(x + 7)^2$$ $$= 2(x^2 + 14x + 49)$$ $$= 2x^2 + 28x + 98$$ What is the domain for f ∘ g?
1) Start with the domain for g: $$g(x) = x + 7$$ $$\text{Domain:} \hspace{.1em} \{x | x ∈ \mathbb{R}\}$$ 2) Continue to the domain for f ∘ g: $$f(g(x)) = 2x^2 + 28x + 98$$ $$\text{Domain:} \hspace{.1em} \{x | x ∈ \mathbb{R}\}$$ Example #3: Find the function f ∘ g and the domain. $$f(x) = \frac{1}{x - 2}$$ $$g(x) = \frac{4}{x}$$ To find f ∘ g, we plug g(x) in for x in f(x): $$(f ∘ g)(x) = f(g(x))$$ $$= f\left(\frac{4}{x}\right)$$ $$=\frac{1}{\frac{4}{x} - 2}$$ $$=\frac{1}{\frac{4}{x} - 2} \cdot \frac{x}{x}$$ $$=\frac{x}{4 - 2x}$$ What is the domain for f ∘ g?
1) Start with the domain for g: $$g(x) = \frac{4}{x}$$ Here x can't be zero since division by zero is undefined. $$\text{Domain:} \hspace{.1em} \{x | x ≠ 0\}$$ 2) Continue to the domain for f ∘ g: $$f(g(x)) = \frac{x}{4 - 2x}$$ Here the denominator (4 - 2x) can't be zero since division by zero is undefined. $$4 - 2x = 0$$ $$x = 2$$ We must also exclude 2 from the domain. $$\text{Domain:} \hspace{.1em} \{x | x ≠ 0, 2\}$$ Example #4: Find the function f ∘ g and the domain. $$f(x) = x^2 - 3$$ $$g(x) = \sqrt{3 - x^2}$$ To find f ∘ g, we plug g(x) in for x in f(x): $$(f ∘ g)(x) = f(g(x))$$ $$= f\left(\sqrt{3 - x^2}\right)$$ $$=(\sqrt{3 - x^2})^2 - 3$$ $$=3 - x^2 - 3$$ $$=-x^2$$ What is the domain for f ∘ g?
1) Start with the domain for g: $$g(x) = \sqrt{3 - x^2}$$ The radicand (3 - x²) must be non-negative. $$3 - x^2 ≥ 0$$ $${-}\sqrt{3} ≤ x ≤ \sqrt{3}$$ $$\text{Domain:} \hspace{.1em} \{x |{-}\sqrt{3} ≤ x ≤ \sqrt{3}\}$$ 2) Continue to the domain for f ∘ g: $$f(g(x)) = -x^2$$ Here, we have no additional restrictions. $$\text{Domain:} \hspace{.1em} \{x |{-}\sqrt{3} ≤ x ≤ \sqrt{3}\}$$

Evaluating Composite Functions

Evaluating a composite function is the process of calculating the specific output of that composite function for a given input. Let's look at an example.
Example #5: Find (f ∘ g)(2). $$f(x) = \frac{2}{x - 1}$$ $$g(x) = x^2$$ We want to find f(g(2)).
Find g(2) first: $$g(2) = 2^2 = 4$$ Now find f(4): $$f(4) = \frac{2}{4 - 1} = \frac{2}{3}$$ $$f(g(2)) = \frac{2}{3}$$ An alternative approach would be to find f(g(x)) first and then plug in a 2 for x. The result is the same. $$f(g(x)) = \frac{2}{x^2 - 1}$$ $$f(g(2)) = \frac{2}{4 - 1} = \frac{2}{3}$$

Decomposing Functions

So far, we have seen that when forming a composite function, we "compose" two functions to form a new function. In some cases, we will want to reverse this process. This involves "decomposing" a function, breaking it down into simpler functions expressed as a composition of two functions. We will see this process used often when studying Calculus. Let's look at an example.
Example #6: Write the function given by f ∘ g as a composition of two functions. $$f(g(x)) = 9x^2 - 12x + 4$$ While various methods can be employed for this task, a commonly intuitive approach often comes to mind initially. Let's factor: $$f(g(x)) = (3x - 2)^2$$ So one way to answer would be: $$f(x) = x^2$$ $$g(x) = 3x - 2$$ Although there are other answers that would be acceptable. For example, we could write our f(g(x)) in a different way. $$f(g(x)) = (3x - 2)^2$$ $$=\left[3\left(x - \frac{2}{3}\right)\right]^2$$ $$=3^2\left(x - \frac{2}{3}\right)^2$$ $$=9\left(x - \frac{2}{3}\right)^2$$ So another way to answer would be: $$f(x) = 9x^2$$ $$g(x) = x - \frac{2}{3}$$

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