domain of 10+3/2x-1

Question

how to find domain and range from an expression given 

 

 

Answer 1

Part 1: Domain questions generally involve any or all of the following three functions:

 

1) A denominator: we know from algebra that denominators cannot equal 0. Any value of x that will result in a denominator equaling 0 is therefore not part of the domain. If your function f(x) = 1/(x-3), we know x can't be 3, but it CAN be anything else.

 

2) A square root: again, from algebra, we know that square roots cannot be negative. Any value of x that will result in the term of a square root becoming negative is not part of the domain. A square root IS, however, allowed to be 0. If your function f(x) = sqrt(x-3), then x must be greater than OR EQUAL TO 3.

 

3) A logarithm: the term in a logarithm can't be negative OR zero. Any value of x that will result in the term in a logarithm being equal to or less than zero is not part of the domain. If your function f(x) = ln(x-3), x must be greater than BUT NOT EQUAL TO 3.

 

Your function only has a denominator for you to worry about. We know that 2x - 1 cannot equal 0. So which value of x makes that result occur?

 

Set an equation and solve for x:

 

2x - 1 = 0

x = 1/2

 

So your domain is all real numbers except for x = 1/2.

 

Part 2: Ranges are a bit more complicated since you need to know how functions behave.

 

For example...

-We know that f(x) = sqrt(x) cannot be negative since square roots always spit out positive or 0 values.

-We know that f(x) = x^2 cannot be negative either since squares always spit out positive or 0 values.

-f(x) = x^3 or f(x) = x or f(x) = 1/x have ranges of all real numbers because their y-values aren't restricted by anything.

 

In your case, since this is a rational function, the range is all real numbers.