What values, if any, are excluded from the domain of the function?

Question

1. f (x) = 3x + 2

3. G (x) = x² + 1

7. H (x) = x² - x + 1

9. f (x) = (2x - 5) / 3

13. F (x) = (x - 2) / 2

 

My math book, in the answer sheet says that "none" is the answer for all these problems; but I have an answer for all of them. Can you please explain why none are excluded from the domain of the function?

 

 

Answer 1

With functions domains, I think it is helpful to consider what you CANNOT use (undefined or answers that are not real). For most functions (there are specific exceptions, see below) there are two main concerns:

 

 

1. you cannot have negative under a square root (or any "even radical")

2. you cannot have zero in the denominator

 

 

1.so for each function: ask yourself "Is there a square root (or other even radical)?" if yes, determine the values of x that make the inside of the radical negative. These value must be excluded from domain

 

    ex.: f(x)=√(x+1) , if x is less than -1 the part under radical is negative, so we can only use value greater than or equal to -1

 

Next, ask yourself "are there any fractions?" if yes, determine what values of x will give you a zero in the denominator. Since you can't have zero in the denominator (undefined) those value must be excluded from the domain.

 

ex.: f(x)=(x+3)/(x-5) , so the denominator is (x-5) so (x-5)≠0  therefore x≠5. 5 must be excluded form the domain.

 

if you don't have either of these "problems" your domain is All Real Numbers* (nothing is excluded)

 

* there are exceptions such as logarithms and some trig functions

 

 

***It is also helpful to note that polynomials always have a domain of: All Real Numbers***

 

Answer 2

The question is asking for what numbers, when you substitute them into x, fail to give a real number as an answer. In each case above, the function given is a polynomial, which you can plug any number into, and get a real number out.

 

Some examples of functions that don't always work are roots and rational functions. For example,

 

S(x) = √(x+1)

You cannot take the square root of a negative number and get a real number back. Therefore, x+1 must be ≥ 0 (non-negative), and x ≥ -1

 

R(x) = 4 / (x-3)

You cannot divide by zero and get a real number. Therefore, x-3 cannot equal zero, and x ≠ 3

 

Please comment if you're still confused.