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## Class 11 Physics (India)

### Course: Class 11 Physics (India)>Unit 3

Lesson 8: Evaluating functions

# What is a function?

Functions assign a single output for each of their inputs. In this video, we see examples of various kinds of functions. Created by Sal Khan.

## Want to join the conversation?

• at why is y a square root of three? why not 3 squared? •   You have to remember that in algebra, what is done to one side of the equation has to also be done to the other side of the equation. When y^2 = 3, in order to find out what y is equal to, you have to get rid of the square. If you square 3, you also have to square the other side of the equation to make it equal. 3^2 would be equal to y^4, which doesn't really help us. Instead, get rid of the square by getting the square root of y squared (which is equal to y) and then finding the square root of the 3.
• So, one day, I asked my dad if a function could be graphed as a circle. He said yes. But I said no because I thought there would be more than one output of the input. For example, see this program I made:
My dad said yes because he said you could find the absolute value of both sides, but I didn't think of it that way. Can someone tell me: who is correct? •   Ωπ fαz919 πΩ ,

By definition of a function, a circle cannot be a solution to a function.
A function, by definition, can only have one output value for any input value. So this is one of the few times your Dad may be incorrect. A circle can be defined by an equation, but the equation is not a function.

But a circle can be graphed by two functions on the same graph.
y=√(r²-x²) and y=-√(r²-x²)

If you look at this program, you will see that it used two functions to create the graph: http://www.khanacademy.org/cs/xr-yr-1/1807411349
• What's the difference between functions in algebra and functions in programming languages? Is here one? •  They are very similar. I think the best way to put it is that a math function takes in some mathematical construct (be it a number or some other variable or even another function) and spits out a mathematical construct, usually a value.
In programming, a function takes in some construct that is defined by the programming language (numbers, strings, classes, the results of another function) and returns a construct defined by the language.
• I don't get how Sal got h(2)=3 and h(8)=11. someone help please? • why are there two variables? •  If you mean, "why does f(x) contain two variables?", please note the f is not a variable. The f is just a way for you to know that when you see f(x) to treat it as a function and not mistakenly treat it as multiplying one variable by the other (it DOES NOT mean f multiplied by x). It does not have to be an f, it can be any symbol and using different symbols such as h(a) helps differentiate one function from another.
• Can a function have multiple inputs? If so, how would you graph said function? • Yes, a function can have multiple inputs. We can graph in the coordinate plane when we have 1 input to 1 output. If we have a function with 2 inputs to create 1 output, we can graph in a 3 dimensional graph of (x, y, z). Once you go to even higher inputs, we typically would not graph them as we don't what a 4-dimensional space looks like.
• Hi! I was wondering if there is a relationship between the equation describing how to plot a circle () and the Pythagorean theorem. • Yep, there definitely is.

Let's consider a circle with center (0, 0) (to make the explanation a little simpler) and radius 3. Let's find some points on the outer edge of the circle.

A noticeable one is (3, 0) (3 units away from the center). Let's try to make a right triangle, where the center of the circle is one vertex, and its opposite vertex is the outer edge. Since this is a right triangle, we should be able to apply the Pythagorean theorem. The base of the triangle would be the x-axis, and the adjacent side would be some y-value. The hypotenuse would be the radius of our circle. Thus, a = x, b = y, and c = r. Using this in the Pythagorean theorem, we find:
x² + y² = r²

Does this work for the point we selected (aka (3, 0))?
3² + 0² = 3² → 9 + 0 = 9 → 9 = 9 ✓

You will find that this works for every single point on the circle. For example, another point on our circle is (3/√2, 3/√2). Does this work in our equation?
(3/√2)² + (3/√2)² = 3² → (9/2) + (9/2) = 9 → 18/2 = 9 → 9 = 9 ✓

When you use this equation with every possible x-value and y-value and graph the points you are able to make, you will construct a circle.
• What does the f stand for? • Basically, the concept of functions gives us a way to name the whole
process of evaluating a particular expression, so we can talk about
it as a whole. We can compare different functions, discuss their
properties, or actually operate on functions to make new functions.
It also broadens the concept, because not all functions can be
written as a simple expression. These two processes, naming things
and extending them, are central to what mathematics is all about.

For example, the first function you showed can be called 'squaring',
and the second can be called 'adding 3'; but most functions would
have to have much more complicated names. By calling one F and the
other G, we have a simple way to discuss them. Some functions, like
the square root and the absolute value, can't be expressed in terms
of more basic functions, but only by inventing a whole new symbol. In
fact, we like to write the square root as 'sqrt(x)', using function
notation, because we don't have the symbol available in e-mail.

We can also treat these names like variables, where we don't know what
specific functions we are calling f and g, yet we can say general
things about the relation of f and g, proving that something is true
for ANY functions, or at least for any functions of a certain type,
all at once. That is powerful!
• How can we graph the fancy function h(a)? •  