Functions
Function evaluation problems appear on 73% of Algebra 1 assessments, making them essential for student success. When students can confidently substitute values and follow order of operations, they build the foundation for advanced mathematical concepts.
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Why it matters
Functions model real-world relationships that students encounter daily. A cell phone plan charging $25 monthly plus $0.10 per text follows f(x) = 25 + 0.1x, where x represents texts sent. Temperature conversion uses f(C) = 9C/5 + 32 to convert Celsius to Fahrenheit. Business profit functions like P(x) = 15x - 200 help companies determine break-even points at 14 units sold. These applications align with CCSS.8.F standards requiring students to interpret functions in context. The LK20.10 curriculum emphasizes practical problem-solving through algebraic thinking, making function evaluation a cornerstone skill for mathematical literacy in economics, science, and engineering careers.
How to solve functions
Functions — Slope & Intercepts
- A function assigns exactly one output to each input.
- Slope = (y₂ − y₁) / (x₂ − x₁) for any two points.
- x-intercept: set y = 0 and solve for x.
- y-intercept: set x = 0 and solve for y.
Example: Line through (1, 3) and (3, 7): slope = (7−3)/(3−1) = 2.
Worked examples
If f(x) = x + 6, find f(3)
Answer: 9
- Substitute x = 3 → f(3) = 3 + 6 = 9 — Replace x with 3 in the expression.
If f(x) = 3x - 7, find f(3)
Answer: 2
- Substitute x = 3 → f(3) = 3 x 3 - 7 = 9 - 7 = 2 — Multiply first, then add or subtract.
If f(x) = x² + 3, find f(3)
Answer: 12
- Calculate x² → 3² = 9 — 3 times 3 equals 9.
- Add 3 → 9 + 3 = 12 — f(3) = 9 + 3 = 12.
Common mistakes
- ✗Students forget order of operations when evaluating f(x) = 2x + 5 at x = 3, writing f(3) = 2 + 3 + 5 = 10 instead of calculating 2(3) + 5 = 11 first.
- ✗Students incorrectly square only the coefficient in f(x) = x² + 4, writing f(3) = 3 + 4 = 7 instead of f(3) = 9 + 4 = 13.
- ✗Students substitute incorrectly in composition problems, evaluating f(g(2)) by finding g(f(2)) instead of first calculating g(2) = 5, then f(5).
- ✗Students drop negative signs when substituting, evaluating f(-2) = (-2)² + 3 as f(-2) = -4 + 3 = -1 instead of f(-2) = 4 + 3 = 7.
Practice on your own
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