Mathematics
Grade 12
15 min
Composition of functions
Composition of functions
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Evaluate composite functions given their algebraic expressions, graphs, or tables of values.
Determine the algebraic expression for a composite function, such as (f ∘ g)(x).
Rigorously determine the domain and range of a composite function by analyzing the constraints of both the inner and outer functions.
Decompose a complex function into a composition of two or more simpler functions.
Verify if two functions are inverses of each other using the property of function composition.
Connect the concept of function composition to its application in calculus, specifically as the foundation for the Chain Rule for differentiation.
Imagine a store offers a 20% discount on an item that's already on sale for $50 off. Is the final price the same regardless of w...
2
Key Concepts & Vocabulary
TermDefinitionExample
FunctionA mathematical rule that assigns exactly one output value for each unique input value. We often write y = f(x), where x is the input and y is the output.f(x) = 2x + 3 is a function. If the input is x = 4, the unique output is f(4) = 2(4) + 3 = 11.
DomainThe set of all possible input values (x-values) for which a function is defined.For the function g(x) = √x, the domain is [0, ∞) because we cannot take the square root of a negative number in the real number system.
RangeThe set of all possible output values (y-values) that a function can produce.For the function g(x) = √x, the range is [0, ∞) because the output of a principal square root is always non-negative.
Composition of FunctionsThe operation of applying one function to the result of another. The output...
3
Core Formulas
Composition Notation: (f ∘ g)(x)
(f \circ g)(x) = f(g(x))
This is read as 'f composed with g of x'. To evaluate, you must work from the inside out: first, evaluate the inner function g(x), then substitute that entire result into the function f.
Composition Notation: (g ∘ f)(x)
(g \circ f)(x) = g(f(x))
This is read as 'g composed with f of x'. The order is reversed. First, evaluate the inner function f(x), then substitute that result into the function g. Note that (f ∘ g)(x) is generally NOT equal to (g ∘ f)(x).
Domain of a Composite Function (f ∘ g)(x)
D_{f \circ g} = \{ x \in D_g \mid g(x) \in D_f \}
The domain of (f ∘ g)(x) consists of all x-values in the domain of the inner function g, for which the output g(x) is in the domain of the outer fun...
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
What is the domain of the function h(x) = ln(√(x² - 9))?
A.(-∞, -3) U (3, ∞)
B.(3, ∞)
C.[-3, 3]
D.(-∞, -3] U [3, ∞)
Challenging
Given f(x) = x² - 4 and g(x) = 2x + 1, find all values of x such that (f ∘ g)(x) = 5.
A.x = 1
B.x = 1 and x = -2
C.x = -1 and x = 2
D.x = 3 and x = -3
Challenging
The domain of f(x) is [-10, 10] and the range of g(x) is [0, 20]. For the composite function (f ∘ g)(x) to be defined for a value x, what condition must be met?
A.x must be in the set [-10, 10]
B.x must be in the domain of g, and g(x) must be in the set [0, 10]
C.x must be in the domain of g, and g(x) must be in the set [-10, 10]
D.The function (f ∘ g)(x) is never defined.
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free