Two quantities that change together can be linked by a function: a rule that takes each input and produces exactly one output. The set of allowed inputs is the domain; the set of outputs that actually occur is the range. The input variable is the independent variable and the output variable is the dependent variable, because its value depends on the input.
Pick a relation. Each arrow sends an input (left) to an output (right). A relation is a function only if no input has more than one arrow leaving it.
Two functions f and g are equal when they have the same domain and give the same output for every input in that domain. So f(x) = x² − 9 and g(x) = (x − 3)(x + 3), both defined for all real numbers, are equal. But h(x) = (x² − 9)/(x − 3) is not equal to f, because 3 is not in the domain of h.
The input and output values vary in tandem according to the function rule, and that rule can be shown graphically, numerically, analytically (with an equation), or verbally. Two ideas let you move through any of them:
| Term | Meaning | How many? |
|---|---|---|
| Image of a | the output f(a) for the input a | exactly one |
| Preimage of k | the set of inputs x with f(x) = k | none, one, or many |
Here f(x) = x² − 2x − 3. Slide the input a to see its image. Slide the output k to see its preimage, the input values that produce k. The preimage can have two values, one value, or none.
To describe how a function behaves, compare function values for inputs in order. A function is increasing over an interval if, as the input values increase, the output values always increase. It is decreasing if the output values always decrease.
The word alwaysmatters: the statement has to be true for every pair a < b in the interval, not just the endpoints.
f(x) = x³ − 3x. Drag both inputs to choose an interval. The highlighted piece of the graph is the interval. A function is increasing there only if the outputs always go up as the inputs go up.
The graph of a function displays a set of input–output pairs and shows how the input and output values vary. Each point (a, f(a)) is one pair. Reading left to right shows what the output does as the input increases.
A verbal description of how two things change together can be the basis for building a graph. Choose which quantity is the input (usually time), then translate each phrase into a shape: rises means the graph goes up, stays the same means flat, falls means down, and faster or slower means steeper or gentler.
A verbal description can be the basis for a graph. Pick a story, then click each part of the story to see which piece of the graph it controls.
Increasing and decreasing tell you the direction of the outputs. Concavity tells you how the rate of change is itself changing.
| Graph is… | When the rate of change is… | The picture |
|---|---|---|
| concave up | increasing | bends like a cup, ∪ |
| concave down | decreasing | bends like a cap, ∩ |
Pick a function. The dashed segments cover equal-length input intervals (width 1). Their slopes are the average rates of change. If the rates of change are increasing the graph is concave up; if they are decreasing it is concave down.
The graph meets the x-axis exactly when the output value is zero. The corresponding input values are called the zeros of the function. A zero is a preimage: the preimage of 0.
f(x) = x² − 2x − 3 + s. Slide s to move the graph up and down. A zero is an input whose output is 0, so zeros are where the graph meets the x-axis.