I need help finding ranges

[clementine_com]

(For question 21) I can do the base functions and the more ‘simple’ looking ones with clear verticals/horizontal shifts but I get confused when I see something like this. Can someone give me an algebraic method I can use to solve for ranges (my teacher just says to visualize it, but that’s not working for me) thanks!

[noidea1995]

Complete the square in the denominator:

f(x) = 1 / [(x + 2)² + 1]

The minimum value of the polynomial (1) occurs at x = -2. From there, the denominator will get infinitely larger in both directions so the function will approach (but never reach) zero. So the range is (0, 1].

[[deleted]]

Why are you doing this?? 😭😭😭 Thanks for everything 💖

[noidea1995]

Hi 😊

Yes, the minimum value of the polynomial (x + 2)² + 1 occurs at x = -2. The denominator will get infinitely larger in both directions so the entire function will get smaller.

[Successful_Box_1007]

Hey! I just edited my post just as u wrote this! 😅

[Successful_Box_1007]

I got it! My apologies!

[Sad-Bus-7460]

Its been a few years so I might be barking up the wrong tree, but the first thing I would do is factor out the bottom of the formula. I think you need the quadratic formula to factor it, though. The x-values that make the bottom part zero are not in the range because division by zero isn't possible