Question

How do you find the range of a function?

I know how to find the domain, but how do you find the range of a function algebraically without a graph? Please don't give me an easy example and solve it. Instead what method should I use to get a reliable answer every time?

Answer 1

you can find horizontal asymptotes, determine what happens as x approaches infinity and negative infinity

Answer 2

when you take calculus you'll be able to find absolute max and absolute min

Answer 3

\[f(x) = \sqrt{-x^2 - x + 6}\] How do you find the range of that? I know the domain is x is not -3, 2

Answer 4

that's not the correct domain

Answer 5

you have the critical numbers though, which is a start

Answer 6

you found the zeros of -x^2-x+6

Answer 7

but -x^2-x+6 can't be negative

Answer 8

do you know how to test for that?

Answer 9

I don't

Answer 10

ok the zeros you found divide the x-axis into intervals

Answer 11

they are the x-intercepts obviously

Answer 12

so the intervals you are to consider are these:

Answer 13

\[(-\infty, -3), (-3, 2), (2, \infty)\]

Answer 14

Ok. How do you find the domain from that though?

Answer 15

now we pick a random number from these intervals and check the sign of f(x) in that interval. all numbers in the interval will have the same sign

Answer 16

we can reconstruct the polynomial using the zeros you found

Answer 17

\[(x+3)(x-2)\ge 0\]

Answer 18

so now lets see what happens when we pick a value from the interval (-infinity, -3)

Answer 19

i'll use -4 for this example

Answer 20

(-4+3)(-4-2)= a positive number, so its greater than 0

Answer 21

negative*negative=positive

Answer 22

Yep

Answer 23

right, so now you know how to do that, we can use a number line