# Functions

Function notation can be used to make meaningful connections between expressions, tables, and other mathematical objects. Use function notation to evaluate a function at a specific value, attach a moveable point to a curve, create input and output tables, and more. Get started with the video on the right, then dive deeper with the resources and challenges below.

## Function Notation

Evaluate functions for a certain value using function notation. For example, if you set $$f(x) = x^2+3x$$, you can then evaluate $$f(3) = 18$$. You can also define a function with more than one variable. Try typing $$g(x,y) = x^2+3y$$ and note that $$g(10,3)$$ evaluates to $$10^2+3(3)=109$$. Click on any of the images to open the graph and explore.

Once a function has been defined, it can be used in expressions or within other functions. With $$f(x) = x^2 + 3x$$, try adding $$2$$ to the function (performing a vertical shift) by typing $$f(x) + 2$$.

Try using function notation with a slider to trace a function. For example, set $$f(x) = \sin(x)$$, and then type $$(a,f(a))$$ in a new line. Desmos will create an adjustable slider for the variable $$a$$. As you drag the slider, the point will trace along the curve.

## Functions in Geometry and 3D

In the Desmos Geometry Tool, transformations are defined as functions. As an example, trying using the toolbar to define a $$180°$$ rotation. When you pull the transformation down from the token navigator, you will see it defined as a function $$T_1(x) =$$ rotate$$(x$$,point,$$180)$$.

Since transformations are defined as functions, you can use function notation to use them repeatedly. Try applying T1 to a different object or composing multiple transformations like in the example.

Open the example graph by clicking on the GIF to see these transformations on a triangle and try out your own transformations.

You can also apply a function to points! Let’s look at the function $$f(a) = \frac{1}{2}a + (0,0,2)$$. In this example, each point $$(a)$$ is scaled by a factor of $$\frac{1}{2}$$ before it’s translated up $$2$$ units on the $$z$$ axis.

Explore this function by clicking on the image.