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

2 Answers
May 28, 2018

Answer:

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

Answer:

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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