**Trigonometry and Functions **

**I**n this lesson, we study very important and useful mathematical objects called functions. If you plug a number into a function, out will come another number. This can be a handy way to relate pieces of information. We practice plugging numbers into functions. We also figure out which numbers ought not be put into certain functions. At the very end, we look at functions that are evaluated through a process instead of a formula.

**Defining Functions**

**Functions** are a bit like equations and formulas. In fact, if you have ever seen an equation with one variable isolated on one side of the equals sign, such as

y=x^{2}+ 2

then you have seen a function.

Functions relate numbers to other numbers. In the equation *y* = *x*^{2} + 2, when *x* = 3, we can compute that *y* = 3^{2} + 2 = 11. Here, 3 relates to 11.

To evaluate other relationships, we need only plug in other numbers. What does 5 relate to? Because 5^{2} + 2 = 27, the number 5 relates to 27.

Every equation with two variables is going to relate numbers like this. The key to being a function is that each number should relate to only one other. If you want the number 4 to relate to 7 and 12 at the same time, then this relationship cannot be defined by a function.

For example, *y* = ±√25 – *x*2 is not a function. If we want to know what the number *x* = 4 relates to, we plug it in and get *y* = ±√25 – 42 = ±√9 = ± 3. Here, we see that *x* = 4 relates to two different numbers, 3 and – 3. This cannot be a function.

The formula *A* = π*r*^{2} does define a function. If we plug in a number for *r*, only one number will come out. This formula gives the area of a circle with radius *r*. We could not have a circle with two different areas, so this formula is a function.

If we are going to talk about more than one function at a time, like *y* = *x*^{2} + 2 and , then we give them names to identify them. Usually, we pick names that are short and easy to write, like *f* (for "function") and *g* (for "the letter that comes after *f*"). First, we write the name. Next, we represent the number to be plugged in with a variable in parentheses. Finally, we set it equal to the formula that defines the function.

f(x) =x^{2}+ 2

*Tip*

*Parentheses are used to mean many different things in mathematics. In a function, they are used to separate the name of the function from the name of the variable that gets plugged in. This does not mean multiplication, as it did in algebra. In algebra, x(x + 5) can be multiplied out to equal x ^{2} + 5x. With functions, f(x) is a combination of the name of the function with the name of its variable. This has nothing to do with multiplication.*

What does the function g relate to the number 7? Because the function g is written *g*(*x*) , the plug-in variable is *x*. Thus, we replace every *x* with the number 7.

The functiongrelates the number 7 withg(7), which in this case is .

**Example 1**

Suppose *s*(*t*) = 200 + l0*t* – 16*t*^{2} · What number is related to 2?

The first part of the equation *s*(*t*) tells us that *s* is the name of the function and that *t* represents the numbers that get plugged in. The function *s* relates the number 2 with *s*(2). We calculate this by replacing each *t*in the formula with the number 2.

s(2) = 200 + 10 · (2) – 16 · (2)^{2}= 200 + 20 – 16 · 4 = 220 – 64 = 156.

Thus, *s*(*t*) = 200 + l0*t* – 16*t*^{2} relates the number 2 to 156.

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