Function

Sommaire

In short: Function describes a rule that gives each allowed input exactly one output, so the same input can never lead to two different results.

8th GradeFunctions and Graphs8.F.A.1

What is Function?

A function works like a vending machine: press one button and exactly one snack falls. Each input, or x-value, is matched with a single output, or y-value, though two different inputs may share an output.

Notation such as \(f(x) = 3x – 4\) names the rule: put 6 in, and \(f(6) = 14\) comes out. A set of ordered pairs is a function only if no x-value repeats with different y-values. On a graph, any vertical line may touch the curve at most once.

How to spot it

A table, list of pairs or mapping diagram links inputs to outputs, and you must test whether the link is valid.
Notation such as f(x) or g(t) appears with a formula to be evaluated.
A graph is drawn and the question uses the vertical line test.
A story has one quantity completely settling another, such as shipping cost by weight.

The method

List the inputs (the x-values) and look for any repeated input.
If a repeated input has different outputs, the relation is not a function; if every input has one output, it is.
To evaluate, replace the variable in the rule with the given input and compute.
On a graph, slide a vertical line across: more than one intersection at any position means no function.

Every input has exactly one arrow leaving it, so the pairs (1,4), (2,7), (3,10), (5,16) form a function.

Function: solved example

Problem

Is \(\{(1, 4), (2, 7), (3, 10), (5, 16)\}\) a function? Is \(\{(2, 5), (2, 8), (4, 1)\}\)? Then evaluate \(f(x) = 3x – 4\) at \(x = 6\).

Solution

In the first set, the inputs 1, 2, 3, 5 are all different, so each has one output: yes. In the second, the input 2 appears with both 5 and 8: no.

For the rule: \(f(6) = 3(6) – 4 = 14\). So yes, no, and \(f(6) = 14\).

Try it

Question 1

Does a circle drawn on a coordinate plane give a single y for each x?

Show the answer

No. A vertical line through the middle of a circle hits it twice, so one x-value has two y-values.

Question 2

If \(g(x) = x^2 + 1\), find \(g(-3)\).

Show the answer

\(g(-3) = (-3)^2 + 1 = 9 + 1 = 10\).

Function vs. Linear Function

Function is the broad idea: any rule with one output per input, whether its graph is a line, a curve or a set of dots. A linear one is a special kind whose graph is a straight line and whose output changes at a constant rate. Every linear rule qualifies, but a rule such as squaring is not linear.

Linear Function

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Whole dictionary: A–Z · 8th Grade · Functions and Graphs

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