domain and range ? (1 Viewer)

[s_t_a_r1234]

how do u work it out, r there any specifics regarding generic rgaphs - r there any rules

 

[insert-username]

Domain is all possible x values. In general, unless your domain is restricted (i.e. you're told 2 < x < 12, etc), x can be any real number. The exception to the rule is when you have a fraction, e.g. 1/(x-1). In this case, when the denominator is zero, the fraction is undefined, so x cannot be a value that makes the denominator zero.

Range is all possible y-values, and this is dependent to some extent the type of function you have and the domain. If you have a parabola, there will be a minimum y-value: the range of y = x^2 is y greater than or equal to zero. If you have a hyperbola, say, y = 1/x, then y can never equal zero since the fraction will never equal zero. For range, look at the values the function can and can't have. I hope that rambling helps a bit.


I_F

 

[acmilan]

In the HSC, you will encounter three main functions where the natural domain is not all real x values, these are:

- logarithmic:

if y = lna, where a is some function involving x, a has to always be > 0.

Example: If y = log(x² - x), then the domain is x² - x > 0, which, when solved, is x < 0 and x > 1. So the domain of ln(x² - x) is x < 0 and x > 1.

- square root:

If y = sqrt(a), where a is some function of x, then the domain is a > 0.

Example: If y = sqrt(x² - x), then the domain is x² - x > 0, which, like above, is x < 0 and x > 1. So the domain of sqrt(x² - x) is x < 0 and x > 1.

- Fractions:

If y = 1/a, where a is some function of x, then the domain is all x except when a = 0.

Example: y = 1/(x² - x), the domain is all real x except x² - x = 0, ie. all real x except x = 0 and x = 1

Summary:

If a is a function of x, then the domain of:

y = ln(a) is a > 0
y = sqrt(a) is a > 0
y = 1/a is all a except a = 0

I think they are the main functions that dont have domain of all real x.

 

[s_t_a_r1234]

n also how would you find out the domain and range when you have like +1 on the side n what if your fn is y=x^2+x ?

 

[Trev]

y=x²+x
Domain is all real x.
To find the range of y, make x the subject:
(x+1/2)²-1/4=y
x= -1/2 +/-√(1/4 + y)
x must be real, so 1/4+y>=0 so range is y>=-1/4.

 

[s_t_a_r1234]

is that what yiu do to work out the range for any given fn with any given form eg linear, to make x the subject then make it = 0?

 

[Trev]

Yeah, that's how I do it.
You don't make it equal to zero. In that case I said it must be greater than or equal to zero as it is within a square root because it has to have a real domain.