Question

For the function, f(x) = -log(3-x)+4
a) State the domain and range:
b) Solve for the x and y intercepts:
c) List any asymptotes:
d) Sketch and include properties in your sketch:
PLEASE SHOW YOUR WORK :)

Answer 1

Do you know what the domain of the log function is?

Answer 2

I have not a clue on how to do this

Answer 3

Okay, first, it'd be good to know that the log function is defined on the domain (0,∞).

Answer 4

That means that, no matter what, the 3-x inside the log function cannot be neither zero nor negative. In other words, 3-x>0. Can you solve for x?

Answer 5

I'm confused

Answer 6

Solve for x\[3-x>0\]

Answer 7

I haven't learned that yet, I don't think

Answer 8

You have:\[3-x>0.\]Add x to both sides:\[3-x+x>0+x,\]\[3-\not{x}+\not{x}>x,\]\[3>x.\]

Answer 9

If x < 3, do you know what the domain of the function f(x)=-log(3-x)+4 is?

Answer 10

so the domain and range would be...?

Answer 11

x<3?

Answer 12

That means that x cannot be greater than three. In other words, your domain is (-∞,3) or x<3, that's correct.

Answer 13

so i would write it as xer, x<3?

Answer 14

Now, the range is a totally different story. For it, you need to calculate how the function behaves as x→3 and x→-∞.

Answer 15

ok

Answer 16

so how would you do that?

Answer 17

Well, you can evaluate that\[\lim_{x\to3}-\log(3-x)+4=\infty.\]How about as x→-∞?

Answer 18

I'm confused once again.

Answer 19

\[\lim_{x\to-\infty}-\log(3-x)+4=?\]

Answer 20

what's lim?

Answer 21

It's the limit of a function as its independent variable(s) approach a certain/certain number(s).

Answer 22

ok so the range is that y cannot be negative?

Answer 23

I'm so sorry but I have to eat dinner, can you answer all of these questions? I'll be back ASAP.

Answer 24

Well, inspecting\[\lim_{x\to-\infty}-\log(3-x)+4,\]we'll see that 3-x is going to be really huge, log(3-x) is going to be really huge, -log(3-x) is going to be really small, and adding 4 to it won't do anything significant. Thus\[\lim_{x\to-\infty}-\log(3-x)+4=-\infty.\]Since we obtained that the range fits in between −∞ and ∞, then we can conclude that the range of the function is (−∞,∞) or the set of all real numbers\[\mathbb{R}.\]

Answer 25

I'm a little confused: If you couldn't solve a very, very trivial inequality, namely 3 - x > 0, then why are you working problems which require you know what a limit is?

Answer 26

I'm going to give you a few hints.

To find the y-intercept(s), you simplify the following equation:\[y=-\log(3)+4.\]To find the x-intercept(s), you solve the following equation for x:\[0=-\log(3-x)+4.\]Since we found the range of the function to go from -∞ to ∞, then there are no horizontal asymptotes. However, at which point is the function undefined? This point is a vertical asymptote.

Upon knowing all these things, sketching the graph should be a piece of cake.

Have fun!

Answer 27

helloo?