Question

The function
f
is defined by
f
(
x
)
=
(
x
−
5
)
2
−
20
for all real numbers
x
.

What is the range of
f
?

Answer 1

To find the range of f, we need to determine the set of all possible values that f(x) can take on.

If we observe the function f(x) = (x - 5)^2 - 20, we can see that the (x - 5)^2 term will always be greater than or equal to 0, since squaring a real number results in a non-negative value.

So, the smallest value that (x - 5)^2 can take on is 0 when x - 5 equals 0, which means x = 5.

Therefore, f(5) = (5 - 5)^2 - 20 = 0 - 20 = -20.

Since (x - 5)^2 is always non-negative, the range of f(x) is all real numbers less than or equal to -20.

Therefore, the range of f is (-infinity, -20].