How to explain a function to a child
Think of a function like a special machine. You put in a number, the machine follows a rule, and it gives you one answer back. If you put in the same number again, you always get the same answer. A rule that sometimes gives 2 different answers for the same number is not a function!
Worked examples of functions
Example 1 (middle grades): Is this a function?
Suppose a rule says: "Multiply the number by 3, then add 1." In math, that is y = 3x + 1.
- If x = 1, then y = 3(1) + 1 = 3 + 1 = 4.
- If x = 2, then y = 3(2) + 1 = 6 + 1 = 7.
- If x = 5, then y = 3(5) + 1 = 15 + 1 = 16.
Each input x gives exactly one output y, so this rule is a function. Because its equation has the form y = mx + b, this is also a linear function, which graphs as a straight line.
Example 2 (upper grades): A nonlinear function from a real situation
Imagine a ball is thrown upward from a platform. Its height h (in meters) after t seconds is given by the function h(t) = -5t2 + 10t + 2.
- At t = 0, h(0) = -5(0)2 + 10(0) + 2 = 2. The ball starts 2 meters high.
- At t = 1, h(1) = -5(1)2 + 10(1) + 2 = -5 + 10 + 2 = 7.
- At t = 2, h(2) = -5(2)2 + 10(2) + 2 = -5(4) + 20 + 2 = -20 + 20 + 2 = 2.
Each time value t has exactly one height h(t), so this is a function. The graph is a curved parabola, so this is a quadratic function, which is a type of nonlinear function (its graph is not a straight line).
Understanding functions more deeply
As your child moves through algebra, functions become a central idea for describing patterns and relationships. Linear functions, written in the form y = mx + b, model constant rates of change, such as earning the same amount of money for every hour worked.
Nonlinear functions include quadratic functions (with x2), exponential functions (with x in the exponent), absolute value functions, step functions, piecewise-defined functions, trigonometric functions, and more. These functions model situations like area, population growth, distance with changing speed, or repeating cycles.
In later grades, your child will study key features of function graphs: slope and y-intercept for linear functions, asymptotes for rational and exponential functions, end behavior (what happens as x becomes very large or very small), and symmetry for even and odd functions. All of these ideas build on the basic rule that a function gives exactly one output for each input.
Common mistakes with functions
- Mistake: Thinking a relation is a function even when one input has 2 outputs, for example the pairs (2, 3) and (2, 5). Correction: A function cannot assign 2 different outputs to the same input; each x can match only one y.
- Mistake: Believing every function must be a straight line. Correction: Linear functions make straight lines, but many functions are nonlinear curves, such as quadratics, exponential functions, absolute value functions, and step functions.
- Mistake: Misreading a graph and thinking a vertical line that touches the graph more than once still shows a function. Correction: A graph represents a function only if every vertical line hits the graph at most once (this is called the vertical line test).
How functions show up in everyday life
Functions appear whenever one quantity depends on another in a predictable way. The total cost at a store depends on how many items are bought, the distance traveled depends on time and speed, and the height of the sun in the sky depends on the time of day. These are all situations that can be modeled with functions.