Introduction to Functions
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn what a function is, how to use function notation like f(x), and how to evaluate functions by plugging in values for the input.
The Concept
Section titled “The Concept”A function is a rule that takes an input (usually called x) and produces exactly one output. It’s like a machine: put something in, get one thing out. No ambiguity.
Examples of functions:
- f(x) = 2x + 3 → double the input and add 3
- g(x) = x² → square the input
- h(x) = 50 + 15x → 50 dollars base plus 15 dollars per hour (x = hours)
Function notation f(x) means “the output of function f when the input is x.”
To evaluate f(x): substitute the given value for x and compute.
Example: f(x) = 2x + 3 If x = 4 → f(4) = 2(4) + 3 = 8 + 3 = 11
Functions can be written as equations (y = …), tables, graphs, or words. All describe the same rule.
Worked Example
Section titled “Worked Example”Evaluate each:
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f(x) = 3x − 7 when x = 5
f(5) = 3(5) − 7 = 15 − 7 = 8
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g(x) = x² + 1 when x = −3
g(−3) = (−3)² + 1 = 9 + 1 = 10
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Cost function: c(h) = 25 + 12h (25 dollars base + 12 dollars per hour)
For h = 4 hours: c(4) = 25 + 12(4) = 25 + 48 = 73 dollars
Real-World Application
Section titled “Real-World Application”Functions model relationships in daily life:
- Pay: p(h) = 18h (18 dollars per hour × h hours worked)
- Fuel cost: c(g) = 3.80g (3.80 dollars per gallon × g gallons)
- Distance: d(t) = 55t (55 mph × t hours driving)
- Tip calculator: t(b) = 0.15b (15% of bill b dollars)
Functions help predict outcomes, compare options, and plan (e.g., “how much will I earn in 40 hours?”).