find the domain of each function, give ur answers in interval notation, f(x)=x-1/x^2-9 and f(x)= square root of 4-5x

Question

Please make the explanation and steps easy to understand.

Answer 1

This function f(x) = (x-1)/x²-9 is called a rational function, which is like a fraction that has a top (numerator) and a bottom (denominator). Since you can't divide by zero then x² - 9 cannot = 0, or x² cannot = 9.

Therefore x cannot = +3 or -3 because (+3)² = 9 and (-3)² = 9

The domain for this function is the set of all x's between -infinity and +infinity EXCEPT for +3 and -3.

If you make a diagram of this using a number line like this you can see that the domain is split into 3 parts.

<-------------- -3 ----------------- +3 ---------------->

So you write the domain in interval notation in three parts:

1) from negative infinity up to but not including -3, written in interval notation: (-infinity, -3)

2) between -3 and +3, so written in interval notation this is: (-3, 3)

3) from +3 up to positive infinity, so written in interval notation this part is : (3, +infinity)

Note that in interval notation parentheses ( ) means up to but not including the endpoints while brackets [ ] means including the endpoints.

For the domain of f(x) = sqrt (4-5x), all numbers can be substituted for x EXCEPT for any that will make what is under the square root negative (less than zero). That's because you can't take the square root of a negative number (it is called imaginary). That means 4 - 5x cannot be < 0, or that x cannot be > 4/5.

If you diagram this on the number line: <----x is OK here----4/5 --x cannot be here---------------->

So x is less than or equal to 4/5, or x < = 4/5 , which in interval notation would be (-infinity, 4/5].