FORMULAE!!!

Here, i just want to share some of the formula of the sign of trigonometric functions that I know. This formula is just to remember that what quadrant that tangent, cosine,sine, and all functions that are positive region. This chapter is the 10th chapter in Precalculus subject.

After Raya Break

I think at this moment, the difficult chapter is the logarithmic functions.I found hard to understand this chapter because it involve many step to solve the question.
I think, i need to study this chapter more harder. This is because, after raya break, we we'll face for 2nd test.
The 2nd test will be more harder and difficult. Furthermore, there will be more chapter for the 2nd test. So I hope I can score in the 2nd test with flying colours..huhu

Example Graph Of Inverse Functions

This is an example of graph of inverse functions.
The question is g:x--> x^2+x-1, x≥-1/2
The question can be found in Tutorial 5.
We sketch this graph on Maple software during math class.
The line of answer is the red colour, while the answer for the inverse is the blue colour.

INVERSE FUNCTIONS

THE INVERSE of a function undoes the action of that function.



Say, for example, that a function f acts on 5, producing f(5). Then if g is the inverse of f, then g acting on f(5) will bring back 5

g(f(5)) = 5.

Actually, g must do that for all values in the domain of f. And f must do that for all values in the domain of g. Here is the definition:



Functions f(x) and g(x) are inverses of one another if:

f(g(x)) = x and g(f(x)) = x,

for all values in their respective domains.

Example 1. Let f(x) = x + 2, and g(x) = x − 2. Then they are inverses of one another. For g(x), which subtracts 2 from a number, is the inverse of adding 2: f(x).

Formally, according to the definition:

f(g(x)) = f(x − 2) = (x − 2) + 2 = x,

(f adds 2 to its argument), and

g(f(x)) = g(x + 2) = (x + 2) − 2 = x.

(g subtracts 2 from its argument.)

The definition is satisfied.

Problem 1. Let f(x) = x² and g(x) = x½. Show that they are inverses of one another. (The domain of f must be restricted to x 0.)

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

f(g(x)) = f(x½) = (x½)² = x,

and

g(f(x)) = g(x²) = (x²)½ = x.



Constructing the inverse

When we have a function y = f(x) -- for example

y = x²

-- then we can often "invert" the equation by solving for x. In this case,



x now appears as a function of y. Therefore on exchanging the variables,



is the inverse function of y = x².

(Taking the square root of a number is the inverse of squaring a number.)

Hence, to construct the inverse of a function y = f(x):

Solve for x, then exchange the variables.

Example 2. What function is the inverse of y = 3x + 4?

Solution. Exchange the sides of the equation, and solve for x:

3x + 4 = y

3x = y − 4

x = y − 4
3 .

Exchange the variables:

y = x − 4
3 .

That function is the inverse of y = 3x + 4.

FUNCTIONS

WHEN ONE THING DEPENDS on another, as, for example, the area of a circle depends on the radius, or the temperature on the mountain depends on the height, then we say that the first is a "function" of the other. The area of a circle is a function of -- it depends on -- the radius.
Mathematically:
A relationship between two variables, typically x and y, is called a function, if there is a rule that assigns to each value of x one and only one value of y.
Thus a "function" must be single-valued ("one and only one"). For example,
y = 2x + 3.
To each value of x there is a unique value of y.
The values (Topic 2) that x may assume are called the domain of the function. We say those are the values for which the function is defined.
In the function y = 2x + 3, the domain may include all real numbers (Topic 2). x could be any real number. Or, as in Example 1 below, the domain may be arbitrarily restricted.
There is one case however in which the domain must be restricted: A denominator may not be 0. In this function,
y
=
1 x − 2
,
x may not take the value 2. For, division by 0 is an excluded operation. (Lesson 6 of Algebra.)
Once the domain has been defined, then the values of y that correspond to each value of x, are called the range. Thus if 5 is a value in the domain of y = 2x + 3, then y = 2· 5 + 3 = 13 is the corresponding value in the range.
By the value of the function we mean the value of y. Again, when x = 5, we say that the value of the function is 13. The range, then, is composed of the values of the function.
It is customary to call x the independent variable, because we are given, or we must choose, the value of x first. y is then called the dependent variable, because its value will depend on the value of x.
Example 1. Let the domain of a function be this set of values:
A = {0, 1, 2, −2}
and let the variable x assume each value. Let the rule that relates the value of y to the value of x be the following:
y = x² + 1.
a) Write the set of ordered pairs (x, y) which "represents" this function.
Answer. {(0, 1), (1, 2), (2, 5), (−2, 5)}
That is, when x = 0, then y = 0² + 1 = 1.
When x = 1, then y = 1² + 1 = 2. And so on.
b) Write the set B which is the range of the function.
Answer. B = {1, 2, 5, 5}. The values in the range are simply those values of y that correspond to each value of x.
Notice that to each value of x in the domain there corresponds one -- and only one -- value of the function. Even though the value 5 is repeated, it is still one and only one value.
Example 2. Here is a relationship in which y is not a function of x:
y² = x
When x = 4, for example -- y² = 4 -- then y = 2 or −2. To each value of x, there is more than one value of y.
Problem 1. Let y be a function of x as follows:
y = 3x²