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§ Linear Alg

Functions

§ Linear Alg

Functions

CCSS.8.F3 min read

A function is a mathematical relationship that assigns exactly one output value to each input value. Functions are typically written in the form f(x) = expression, where x represents the input and the expression shows how to calculate the output. For example, f(x) = 2x + 3 means multiply the input by 2 and add 3.

§ 01

Why it matters

Functions model countless real-world relationships where one quantity depends on another. Temperature conversion formulas like F = 95C + 32 are functions that convert Celsius to Fahrenheit. Business profit models use functions to relate revenue to the number of items sold — if each item generates $15 profit, then P(x) = 15x represents total profit from x items. Population growth, medication dosage calculations, and compound interest formulas all rely on functions. In advanced mathematics, functions become the foundation for calculus, where students analyze rates of change and area under curves. The CCSS Grade 8 standards (CCSS.8.F) introduce function evaluation and comparison, preparing students for Algebra I function notation and eventually pre-calculus analysis of polynomial, rational, exponential, and logarithmic functions.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

If f(x) = x + 5, find f(3)

Answer: 8

  1. Substitute x = 3 → f(3) = 3 + 5 = 8 — Replace x with 3 in the expression.
Easy§ 02

If f(x) = 5x - 5, find f(3)

Answer: 10

  1. Substitute x = 3 → f(3) = 5 x 3 - 5 = 15 - 5 = 10 — Multiply first, then add or subtract.
Medium§ 03

If f(x) = x² + 5, find f(6)

Answer: 41

  1. Calculate x² → 6² = 36 — 6 times 6 equals 36.
  2. Add 5 → 36 + 5 = 41 — f(6) = 36 + 5 = 41.
§ 04

Common mistakes

  • Writing f(3) = x + 5 instead of f(3) = 3 + 5 = 8 when evaluating f(x) = x + 5, failing to substitute the input value for x.
  • Computing f(4) = 2 × 4 + 3 = 8 + 3 = 11 as f(4) = 2 × 4 + 3 = 6 + 3 = 9, forgetting order of operations by adding before multiplying.
  • Confusing function composition f(g(2)) by calculating f(2) and g(2) separately instead of first finding g(2) = 7, then computing f(7).
§ 05

Frequently asked questions

What is the difference between f(x) and f(3)?
f(x) represents the general function rule using variable x, while f(3) is the specific output when x equals 3. If f(x) = x + 5, then f(3) = 3 + 5 = 8. The notation f(3) means evaluate the function at the input value 3.
How do you evaluate a function step by step?
Replace every instance of the input variable with the given number, then calculate following order of operations. For f(x) = 3x² - 2 and input x = 4: substitute to get 3(4)² - 2, calculate 4² = 16, multiply 3 × 16 = 48, subtract to get 48 - 2 = 46.
What does function notation f(x) actually mean?
The notation f(x) reads as "f of x" and represents a function named f that takes input x. The letter f is just the function's name — it could be g(x), h(x), or any letter. The parentheses don't mean multiplication; they indicate that x is the input to function f.
Can a function have the same output for different inputs?
Yes, multiple inputs can produce the same output. For example, f(x) = x² gives f(3) = 9 and f(-3) = 9. However, each input must have exactly one output. A function cannot assign both 5 and 7 to the same input value.
How do you find the slope and intercepts of a linear function?
For f(x) = mx + b, the slope is m and y-intercept is b. To find x-intercept, set f(x) = 0 and solve for x. For f(x) = 2x - 6: slope is 2, y-intercept is -6, and x-intercept is found by solving 2x - 6 = 0, giving x = 3.
§ 06

See also

FunctionExpressionAreaPolynomialSlopeSetVariableMeanMultiple
§ 06

Where to next?

Prerequisites
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