Give an example of a function, f(x), with a domain of (0,5] and a range of [0,∞).
I've came up with the following. Is this right or do I need to change something?
|(1/x) + ((1/x) - 5))|
You missed your other 1/x on the other side and youhave to multiply.
Could you please explain what you mean?
Do you mean instead of adding like I am to multiply the first 1/x by the ((1/x) - 5)?
To determine whether the given function, f(x) = |(1/x) + ((1/x) - 5))|, has the correct domain and range, we need to verify if it matches the requirements.
First, let's consider the domain. The domain of a function refers to the set of all possible input values for x. In this case, the domain is given as (0,5]. This means that x can take any value greater than 0 up to and including 5.
In our function, we have x as the denominator in the expression (1/x). Therefore, we need to ensure that x values do not equal 0, as this would result in division by zero, which is undefined. Looking at the given domain, (0,5], there is no mention of excluding 0, so we can conclude that this function is not valid for x = 0. Therefore, the domain should be adjusted to (0,5] \ {0}, which means all values greater than 0 up to and including 5, excluding 0.
Next, let's consider the range. The range of a function refers to the set of all possible output values. In this case, the range is given as [0,∞), which means the function's output can be any value greater than or equal to 0.
Considering our function f(x) = |(1/x) + ((1/x) - 5))|, we can examine how the different parts of the expression contribute to the range. Since (1/x) represents the reciprocal of x, it will always be positive (or zero) for all valid values of x in its domain. Then, ((1/x) - 5) will be negative or zero for all values in the domain, excluding x = 0.
Given that the function involves taking the absolute value of the sum of two terms, we can conclude that f(x) will always be positive or zero. Therefore, the range [0,∞) is satisfied.
In conclusion, the given function, f(x) = |(1/x) + ((1/x) - 5))|, with a domain of (0,5] and a range of [0,∞), is valid and matches the specified conditions.