Each x-value has only 1 y-value that it maps to?

Question

Domain: {- infinity, infinity}
Range: {- infinity, 0]

Answer 1

Yes, in the context of a function, the statement "each x-value has only 1 y-value that it maps to" is fundamental to the definition of a function. This means that for every input (x-value), there is exactly one output (y-value).

In your given domain of \((-∞, ∞)\) and a range of \((-∞, 0]\), it suggests that the function takes any real number as an input and produces an output that is either negative or zero.

For example, a common function that fits this description is \( f(x) = -|x| \), which maps any real input x to a non-positive output.

Here’s a breakdown of your parameters:

• Domain: \((-∞, ∞)\) means any real number can be used as an input.
• Range: \((-∞, 0]\) means that the output will always be a real number less than or equal to 0.

As long as each input x corresponds to one unique output y within these specified domains and ranges, the relationship you're describing can be considered a function.

Answer 2

Ok, so what does it map to?

Answer 3

To determine what the function maps to given the domain of \((-∞, ∞)\) and range of \((-∞, 0]\), we need to define a specific function. Since the output must always be less than or equal to 0 for any real input, the function can be of various forms. Here are a few examples of functions that meet those criteria:

1. Linear Function:

   • \(f(x) = -|x|\)
     • This function maps any real number \(x\) to a non-positive value. For example:
       • \(f(1) = -1\)
       • \(f(0) = 0\)
       • \(f(-2) = -2\)

2. Quadratic Function:

   • \(f(x) = -x^2\)
     • This function also maps \(x\) to values in the range \((-∞, 0]\).
       • \(f(1) = -1\)
       • \(f(0) = 0\)
       • \(f(-2) = -4\)

3. Exponential Function:

   • \(f(x) = -e^{-x}\)
     • This is another example, and it maps all real numbers to negative values plus zero.
       • \(f(0) = -1\)
       • \(f(1) = -\frac{1}{e}\)
       • \(f(-1) = -e\)

4. Constant Function:

   • \(f(x) = c\) where \(c \le 0\) (e.g., \(f(x) = -5\))
     • In this case, every input maps to a constant output that is less than or equal to zero.

In summary, without a specific function defined, we can say that the function can take any real number as input and maps it to any value in the range of \((-∞, 0]\). Each input \(x\) will produce exactly one output \(y\) in that range.