Question

How do you find the range of functions? Please use an example.

Answer 1

$$
f(x)=(x+1)^2-2
$$
Since \((x+1)^2\ge0\), what is the lowest value that \(f(x)\) can take?

Answer 2

-1?

Answer 3

$$
(x+1)^2\ge 0
$$
So
$$
f(x) \ge 0-2\\
f(x)\ge-2
$$
Do you see this?

Answer 4

The range of f(x) means what is the lower and upper bound for all values that f(x) can take on

Answer 5

Yes I see that now.

Answer 6

So the lowest value f(x) can take is -2

What is the highest?

Answer 7

The highest is -1 right?

Answer 8

Pick any number greater than -2, say 100, can
$$
(x+1)^2-2>100
$$
This means that
$$
(x+1)^2>102\\
$$
or
$$
x+1 > \sqrt{102}\\
x> \sqrt{102}-1
$$
Can x be greater than \(\sqrt{102}-1\)
yes it can.

If you take any number other than 100, say N then we require that
$$
x> \sqrt{N+2}-1
$$
And x can certainly be bigger than this for any N

Therefore, there is no limit to how high f(x) can be. The upper limit for f(x) in infinity.

Answer 9

range of f(x) then is -2 to infinity

Answer 10

So it would be y is greater than or equal to -2?

Answer 11

yes!

Answer 12

Thank you!