Question

How do you find the domain and range of `2(x-3)`?

Answer 1

Domain: `(-∞,∞)` Range: `(-∞,∞)`

Explanation:

The domain is all of the values of `x` for which the function exists. This function exists for all values of `x`, as it's a linear function; there is no value of `x` which would cause division by `0` or a vertical asymptote, a negative even root, a negative logarithm, or any situation which would cause the function to not exist. The domain is `(-∞,∞)`.

The range is the values of `y` for which the function exists, in other words, the set of all possible resulting `y` values obtained after plugging in `x`. By default, the range of a linear function whose domain is `(-∞,∞)` is `(-∞,∞)`. If we can plug in any `x` value, we can obtain any `y` value.

Answer 2

`x in R`- x can take any real value
`y in R`- y can take any real value

Explanation:

If you image the function as `y=2(x-3)` we can model it as a graph, which should make it more clear.

From the graph we can see that both x and y go on towards infinity, which means that it stretches through all values of x and all values of y, and the fractions of it.

Domain is about: "Which x values can or cannot my function take?" and Range is the same but for the y values the function can or cannot take. However, from the graph we can see that all real values are acceptable answers.

graph{y=2(x-3) [-10, 10, -5, 5]}

Answer 3

Because there are no x values for which a y value does not exist, the domain is all real numbers. The range is also all real numbers.

Explanation:

The domain of a function is all possible x values that encompass the solution set. Discontinuities in the domain come from functions where a domain error is possible, such as rational functions and radical functions.

In a rational function (ex. `5/(x-2)`) the denominator can not be equal to zero. This is because you cannot divide by zero, it produces a domain error. So when stating the domain of this given function, you may use all possible values of x where the denominator does not equal zero (x | x != 2)

In a radical function (ex. `sqrt(x+4)`) the contents inside the square root cannot be equal to a negative number. This is because there are no real positive numbers which multiplied by themselves is equal to a negative number. Therefore, the domain of the function is all possible values of x where the root is positive (x | x>=-4).
(note: for radical functions with an odd root, such as cube roots or 5th roots, negative numbers are within the solution set)

There are other functions which can produce domain errors, but for algebra, these two are the most common.

The range of a function is all the possible y values, to find these it is useful to look at the graph of a function.

Looking at the graph of `x^2`, we can see that as the x values stretch to infinity, there are no negative y values. In other words, the graph never dips below the line y = 0. The range for this function is y | y >= 0)

If you are unsure of the range of a function, the best way to tell is to look at the graph and see the upper and lower limits of the y values.