Mississippi Math A3.A.18

MathGrades 9–12Algebra (A) Analyze and Manipulate Functions

The Standard

Find the composite of two given functions and find the inverse of a given function. Extend this concept to discuss the identity function f(x) = x.

2025 Mississippi College- and Career-Readiness Standards for Mathematics · Perform Operations on Expressions, Equations, Inequalities, and Polynomials

Teacher's field guide

What This Standard Means

What Students Need to Do: Students put one function’s output into another and keep track of the order. They find inverse rules when possible and check them by composing the functions. They recognize that the rule f(x) = x leaves every input unchanged.
What Mastery Looks Like: Given f(x) = 2x + 3 and g(x) = x², a student finds f(g(x)) = 2x² + 3 and g(f(x)) = (2x + 3)². They find f⁻¹(x) = (x − 3)/2 and show that f(f⁻¹(x)) = x. They know x² needs a restricted domain to have an inverse function.
Common Misconceptions: Students may treat f(g(x)) and g(f(x)) as the same, or multiply the functions instead of composing them. They may read f⁻¹(x) as 1/f(x), or assume x² has an inverse function on all real numbers.
How to Assess It: Exit ticket: Let f(x) = 2x + 3 and g(x) = x². Find f(g(x)), find f⁻¹(x), and simplify f(f⁻¹(x)). What does the last result tell you?

Lesson moves

Ways to Teach It

Write f(x) = 2x + 3 and g(x) = x² on index cards; pass input 2 through the cards in both orders.
Ask students to explain in writing why f(g(2)) and g(f(2)) give different results for the two card rules.
In pairs, match function cards such as 3x − 5 with inverse cards, then check each match by composing to get x.
For a taxi charging $3 plus $2 per mile, write a fare rule and its inverse, then use both for a 7-mile ride.

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