Question

How do you find the domain and range of `f(x)=10-x^2`?

Answer 1

Domain = Real Number `( RR )`
Range = `(-oo,10]`

Explanation:

As `x` can take any value so domain is real number.
For range We know that
`x^2>=0`
So
`-x^2<=0`
now add 10 on both side of equation
so equation become
`10-x^2<=10+0`
So the range is `(-oo,10]`

Answer 2

Domain: `x in RR`
Range: `f(x) in (-∞, 10]`

Explanation:

Well, first, let's explain what a domain and range is.

A domain is the set of argument values (or "input") in which the function is defined. So, for example. for a function `g(x) = sqrt(x)`, the domain will be all non-negative real numbers, or `x >= 0`.

For this function `f(x)`, we see that the function has no square roots, fractions, or logarithmic functions that would be undefined for certain values of `x`.

Therefore, the domain of this function is all real numbers, or `x in RR`.

The range of a function is all possible values (or "output") of the function, after substituting in the domain. So, for example, a function such as `h(x) = x` will have the range as all real numbers, but a function such as `j(x) = sin(x)` can only output values in between -1 and 1, so the range is `[-1,1]`, or `-1 <= j(x) <= 1`.

To find the range of `f(x)`, we must first observe that the function has no minimum value. This can be done two ways.

First, we can observe that the coefficient in front of the `x^2` term is negative. So, as `x` increases (or decreases), `x^2` increases, and the value of `f(x)` decreases. Thus there must be a maximum value for `f(x)`, which is 10 in this case, when `x = 0`. You may have to complete the square, or use some other method for other functions.

Or, we can just see the graph of `y = f(x)`. graph{y = 10-x^2}
From the graph, it is clear that the maximum value of `f(x)` is 10.

So, we can conclude that the domain of the function is all real numbers, or `RR`, and the range of the function is `(-∞, 10]` in set notation.

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