What is the range of the function #f(x)=10-x^2#?

1 Answer
Jul 22, 2016

Answer:

#y in (-oo, 10]#

Explanation:

The range of a function represents all the possible output values that you can get by plugging in all the possible #x# values allowed by the function's domain.

In this case, you have no restriction on the domain of the function, meaning that #x# can take any value in #RR#.

Now, the square root of a number is always a positive number when working in #RR#. This means that regardless of the value of #x#, which can take any negative values or any positive value, including #0#, the term #x^2# will always be positive.

#color(purple)(|bar(ul(color(white)(a/a)color(black)(x^2 >=0 color(white)(a)(AA) x in RR)color(white)(a/a)|)))#

This means that the term

#10 - x^2#

will always be smaller than or equal to #10#. It will be smaller than #10# for any #x in RR "\"{0}# and equal to #10# for #x=0#.

The range of the function will thus be

#color(green)(|bar(ul(color(white)(a/a)color(black)(y in (- oo, 10]color(white)(a/a)|)))#

graph{10 - x^2 [-10, 10, -15, 15]}