How do you find the domain and range of f(x)=10-x^2f(x)=10x2?

2 Answers
May 28, 2018

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]

May 28, 2018

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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