1 Finding the domain and range of a function.

1.1 Domain

Domain of a function: set of all allowable values of the independent variable(usually x)

-Question: What can you plug into x so that the function is still defined?

TIPS in finding the domain:1. Always start by assuming that the function has the domain D: (-, )then check for the following two conditions.

2. If plugging in a certain value of x leads to the either of the followingconditions, then that certain value should not be included in the domain.a. The x value makes the denominator of the function 0.b. The x value makes the argument inside a radical sign of even root

negative.

3. Remember that BOTH of these conditions may occur at the same time.

1.2 Range

Range of a function: set of all allowable values of the dependent variable(usually y)

-Question: What are the possible values you can get from the functionwhen you plug the domain into it?

TIPS in finding the domain:1. Always find the domain FIRST.2. If you can and have time, GRAPH the function. This is the safest wayyou can use to determine the range3. Refer to the various cases in the next section.

2 Different forms in finding range

The forms you will probably encounter and what you should do:Linear y = mx + b-The range is always all real numbers.

Quadratic f(x) = ax2 + bx + c

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-Put in the general form of a parabola to find the vertex (h, k)-If a is negative/positive, the vertex is a maximum/minimum. If mini-

mum, then the range is R:[k,). If maximum, then the range is R:(, k]

Absolute Value f(x) = a |g(x)|-If a=0, the range will be all the POSITIVE possible values of g(x).-If a is not 0 and you have +|g(x)|, the range will be R:[a,]-If a is not 0 and you have -|g(x)|, the range will be R:[, a]-NOTE THAT a will just shift the range! This is true for radical and

rational cases as well so I wont list the a cases anymore because they areexactly the same as above.

Radical Value f(x) = cg(x)

-If c=even, range is all positive possible values of g(x) including 0-if c=odd, range is all real numbers-These two are such because when c=even, g(x) cannot be negative and

when c=odd, g(x) can be negative.

Rational f(x) = bg(x)-The range is all real numbers EXCEPT 0 in most cases. This, however,

is different depending on g(x). What you do is first find the range of g(x)(like if it has to be negative or has to be positive) then you adjust accordingly.

3 Exercises

1. y = 3x 12. y = x2 + 33. y = x2 + 24. y = x2 + 4x 15. y =

x 2

6. y = 10 2x7. y = 5x28. y = |x 1|9. y = |x + 23| 110. 32x+4 + 5

11. 2|4x+3| + 3

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