Question

Write the equation of a function with domain and range given, how to do that?

The domain is `{-5<= x <= 5}`
The range is `{0<= y <= 5}`
How to do I determine the equation with these two things given?

Answer 1

`f(x) = sqrt(25-x^2)`

Explanation:

One method is to construct a semicircle of radius `5`, centered at the origin.

The equation for a circle centered at `(x_0, y_0)` with radius `r` is given by `(x-x_0)^2+(y-y_0)^2=r^2`.
Substituting in `(0,0)` and `r=5` we obtain `x^2+y^2=25` or `y^2 = 25-x^2`
Taking the principal root of both sides gives `y = sqrt(25-x^2)`, which fulfills the desired conditions.

graph{sqrt(25-x^2) [-10.29, 9.71, -2.84, 7.16]}

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Note that the above only has a domain of `[-5,5]` if we restrict ourselves to the real numbers `RR`. If we allow for complex numbers `CC`, the domain becomes all of `CC`.

By the same token, however, we can simply define a function with the restricted domain `[-5,5]` and in that manner create infinitely many functions which fulfill the given conditions.

For example, we can define `f` as a function from `[-5,5]` to `RR` where `f(x) = 1/2x+5/2`. Then the domain of `f` is, by definition, `[-5,5]` and the range is `[0,5]`

If we are allowed to restrict our domain, then with a little manipulation, we can construct polynomials of degree `n`, exponential functions, logarithmic functions, trigonometric functions, and others which do not fall into any of those categories, all of which have domain `[-5,5]` and range `[0,5]`