# Defining Functions with lambda

Until now, we have only used built-in Racket procedures. To define our own, we'll use the lambda keyword.

(lambda (<Input1> <Input2> ... <InputN>) <Body>)

; Most of the time, we put <Body> on a new line to make things more readable.
; The following pattern is identical to the previous one, just with a newline.
(lambda (<Input1> <Input2> ... <InputN>)
<Body>)

So we can use lambda to create a function by giving it the following "chunks":

1. A parenthetical sequence of inputs (also known as arguments or parameters), and
2. Some kind of expression, the body of the function.

Here are some quick examples of lambdas and their signatures.

; number -> number
; returns x, doubled
(lambda (x) (* 2 x))

; string -> string
; wishes happy birthday to someone
(lambda (name)
(string-append "Happy birthday, " name "!"))

## Anonymous Functions

Calling lambda creates an anonymous function. By "anonymous", we literally mean "nameless"; our function doesn't have a name we can use to call it.

To understand this more thoroughly, consider the following mathematical function $f$:

$f(x) = 2x$

The Racket equivalent of $f$ is the following:

(define f
(lambda (x) (* 2 x)))

We can decompose each of these function declarations into two chunks: the name, and the lambda:

$\;{\small\begin{matrix}\\ \normalsize f \\ ^{\small \text{name}}\end{matrix} }\; \;{\small\begin{matrix}\\ \normalsize (x) = 2x \\ ^{\small \text{lambda}}\end{matrix} }\;$

(define f  ; name
(lambda (x) (* 2 x)))  ; lambda

Notice that we've added a (define f ... to our lambda.

define attaches a name, f, to our anonymous lambda. Now we can reference our function later on, by calling (f <number>).

To illustrate the importance of what just happened, consider the mathematical version of $f$. If we wanted to call $f$ with a bunch of inputs for $x$, we might write something like

\begin{aligned} f(3) \\ f(4) \\ f(5) \end{aligned}

which is a lot cleaner and more succinct than writing

\begin{aligned} \big( (x) = 2x \big)(3) \\ \big( (x) = 2x \big)(4) \\ \big( (x) = 2x \big)(5) \end{aligned}

To be clear, the above math is perfectly legal. It just looks weird because nobody ever does that.

Notice that although the short and verbose function calls are exactly equivalent, using the function name $f$ gives us a handy way to reference the lambda part.

This logic translates directly to Racket. Here's how we'd call f a bunch of times:

; named version
(define f
(lambda (x) (* 2 x)))

(f 3)  ; 6
(f 4)  ; 8
(f 5)  ; 10

Again, this is much more succinct than calling the lambda anonymously:

; anonymous version
(lambda (x) (* 2 x))

((lambda (x) (* 2 x)) 3)  ; 6
((lambda (x) (* 2 x)) 4)  ; 8
((lambda (x) (* 2 x)) 5)  ; 10

(Once again, writing the whole lambda out each time is cumbersome, but perfectly legal. You can try this in DrRacket and it will work.)

Note: The anonymous function calls are pretty difficult to read. Consider this one:

((lambda (x) (* 2 x)) 3)  ; 6

Remember that the Racket syntax for calling any function is just

(<Function> <Input1> <Input2> ...)

So we call our anonymous lambda the same way:

• <Function> becomes (lambda (x) (* 2 x))
• <Input1> becomes 3 (there is only one <Input>)

and everything is wrapped in a set of parentheses to signal to Racket that we are calling the function.

## Function Evaluation

What actually happens when we call (f 3), anyways? Let's return to our trusty mathematical function for comparison.

1. Look up the full definition of our function.
2. Plug in the input values wherever they appear in the function body.
3. Ignore the "function baggage" and focus on the body, now that we've plugged everything in.
4. Simplify the resulting expression using order of operations, and return the result.
Step Math $f(3)$ Racket (f 3)
Look up definition $(x) = 2x$ (lambda (x) (* 2 x)
Plug in inputs $(3) = 2(3)$ (lambda (3) (* 2 3))
Focus on body $2(3)$ (* 2 3)
Simplify $6$ 6

## More Examples

We used a really simple function to demonstrate the parallels between mathematical function definitions, and Racket function definitions. Here are some more examples.

### Using more expressive names

In math, we're pretty much limited to single-letter function names like $f$ or $g$. In Racket, we can be a lot more descriptive with our naming.

(define doubler
(lambda (x) (* 2 x)))

The same goes for our arguments. While the above function works for doubling numbers generically, suppose we want to write a function designed specifically to double the width of a shape.

(define doubler
(lambda (width) (* 2 width)))  ; much clearer

### Multivariable functions

Just as we have multivariable functions in math, we have multivariable functions in Racket.

$g(x, y, z) = x^2 + y^2 + z$

translates to

(define g
(lambda (x y z)
(+ (sqr x) (sqr y) z)))

(g 1 2 3)  ; 8

The same evaluation pattern works for multivariable functions.

Step Math $g(1, 2, 3)$ Racket (g 1 2 3)
Look up definition of $g$ $(x, y, z) = x^2 + y^2 + z$ (lambda (x y z) (+ (sqr x) (sqr y) z))
Plug in inputs $(1, 2, 3) = (1)^2 + (2)^2 + (3)$ (lambda (1 2 3) (+ (sqr 1) (sqr 2) 3))
Focus on body $(1)^2 + (2)^2 + (3)$ (+ (sqr 1) (sqr 2) 3)
Simplify $1 + 4 + 3 = 8$ (+ 1 4 3) = 8

Here's another example. Using the Pythagorean Theorem, we can write a function to compute the length of the hypotenuse, given two legs of a triangle:

$hypotenuse(x, y) = \sqrt{x^2 + y^2}$

which translates to Racket as

(define hypotenuse
(lambda (x y)
(sqrt (+ (sqr x)
(sqr y)))))

(hypotenuse 3 4)  ; 5
Step Math $hypotenuse(3, 4)$ Racket (hypotenuse 3 4)
Look up definition $(x, y) = \sqrt{x^2 + y^2}$ (lambda (x y) (sqrt (+ (sqr x) (sqr y))))
Plug in inputs $(3, 4) = \sqrt{(3)^2 + (4)^2}$ (lambda (3 4) (sqrt (+ (sqr 3) (sqr 4))))
Focus on body $\sqrt{(3)^2 + (4)^2}$ (sqrt (+ (sqr 3) (sqr 4)))
Simplify $\sqrt{9 + 16} = \sqrt{25} = 5$ (sqrt (+ 9 16)) = (sqrt 25) = 5

### Functions on non-numerical data types

We've mostly looked at functions with the signature

number -> number

so far. But functions can take any inputs and return any outputs. Here are more examples.

; has-even-length? : string -> boolean
; returns true if the given string has an even number of characters
(define has-even-length?
(lambda (str)
(even? (string-length str))))

(has-even-length? "eecs")  ; true
(has-even-length? "lol")  ; false
(has-even-length? "k bye!")  ; true
Step (has-even-length "eecs")
Look up definition (lambda (str) (even? (string-length str)))
Plug in inputs (lambda ("eecs") (even? (string-length "eecs")))
Focus on body (even? (string-length "eecs"))
Simplify (even? 4) = true
; mad-lib : string, string, string -> number
; returns a sentence with the given noun, verb, and adjective
(mad-lib "dignity" "gallop" "new")  ; "I gallop to Tech Express and buy a new dignity."