Lesson 11 of 12 · Algebra Foundations
Functions & Transformations
A function is just a rule; composition is applying two rules in sequence; a transformation shifts or reflects the rule's output. Piecewise functions ask you to pick the right rule based on the input, and functional equations ask you to discover the rule from clues about how it behaves.
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Learning objectives
- Evaluate functions and compositions of functions, working from the inside out.
- Apply and interpret shift, reflection, and stretch transformations.
- Evaluate piecewise functions correctly, and check candidate solutions against the right piece's domain.
- Solve introductory functional equations by substituting strategic values.
The core idea
Composition, transformation, and piecewise definitions are all just ways of building a new function from an old one (or old ones). Composition feeds one function's output into another. A transformation applies a fixed shift, reflection, or stretch to a function's input or output. A piecewise function is really several separate functions, each active on its own domain — evaluating one is just a matter of choosing the right piece first.
Functional equations are different: instead of being given a formula, you're given a rule the function must obey for every input, plus a few specific values. The strategy is almost always the same — substitute clever specific values (0, 1, or values related to what you need) to extract enough information to answer the question.
Composition
(f∘g)(x) = f(g(x))
Always work from the inside out: evaluate the inner function first.
Transformations
f(x)+k shifts the graph up k units. f(x−h) shifts it right h units. −f(x) reflects it over the x-axis. f(−x) reflects it over the y-axis. a·f(x) stretches it vertically by a factor of a.
Functional equations — the standard approach
Substitute specific values for the variables — often 0, 1, equal values, or values that make one side simplify — to generate equations you can solve for the unknown quantities the problem asks about.
Worked example 1 — Composition
Problem
If f(x)=2x+1 and g(x)=x²−3, find f(g(2)).
Key insight
Work from the inside out: find g(2) first.
Solution
g(2)=1, then f(1)=3.
Takeaway
f(g(x)) is read right-to-left in terms of order of operations: g acts first, then f acts on the result.
Worked example 2 — Transformation
Problem
If f(x)=x² and g(x)=f(x−3)+2, find g(5).
Key insight
g shifts f right by 3 and up by 2 — substitute directly rather than trying to visualize the shift.
Solution
f(2)+2 = 4+2 = 6.
Takeaway
When a specific numeric value is needed, direct substitution is faster and safer than reasoning about the graph shift abstractly.
Worked example 3 — Piecewise function
Problem
If f(x)=x+1 for x<2, and f(x)=3x−1 for x≥2, find f(2)+f(0).
Key insight
Check each input against the domain conditions separately before evaluating.
Solution
x=2 uses the second piece (x≥2): f(2)=5. x=0 uses the first piece (x<2): f(0)=1. Sum: 6.
Takeaway
Boundary values like x=2 here are the most common source of piecewise errors — check the inequality's direction (≥ vs >) carefully.
Worked example 4 — Functional equation
Problem
A function satisfies f(x+y)=f(x)+f(y) for all real x, y, and f(1)=3. Find f(5).
Key insight
Write 5 as 1+1+1+1+1 and apply the given equation repeatedly.
Solution
f(5) = 5f(1) = 15.
Takeaway
This "additive" functional equation always gives f(n) = n·f(1) for positive integers n — a pattern worth recognizing on sight.
Strategy notes
Compositions: always work from the inside out
f(g(x)) means "apply g first, then apply f to the result." Reversing this order is one of the most common errors on composition problems.
Functional equations: substitute strategically, not randomly
Look for substitutions that make one side of the equation simplify dramatically — setting a variable to 0, setting two variables equal, or choosing values that directly relate to what the problem asks for.
Common mistakes
Evaluating a composition in the wrong order
f(g(x)) is not the same as g(f(x)) in general. Always identify which function is "outer" and which is "inner" before starting.
Using the wrong piece at a boundary value
When an input falls exactly on the boundary between two pieces (like x=2 in a "x<2 vs x≥2" split), only one piece actually applies — check the inequality symbol carefully, not just which piece "looks closer."
Assuming a functional equation's pattern without verifying it
Patterns like f(n)=n·f(1) hold for specific functional equations (like additive ones), not automatically for every functional equation. Derive the pattern from the given equation rather than assuming it.
Practice
20 questions across four difficulty tiers. Most want a single numeric answer. One piecewise question has two valid solutions — enter both, comma-separated. Up to three tries per question before the solution is shown.