Domain and Range

Question

What is the domain and range of the following ?

1) y=ln(x-4)

2) y=ln(x)-7

3) y=ln(x+sqrt2)

4) y=ln(x-3)+5

5) y=1/(x+8)

6) y=(1/x)+3

7) y=(1/x)-(1/5)

8) y=1/(x-2)-ln7

Answer 1

Sarah,

 

Domain has to to do with x values and range goes with y values

 

For the first equation,

 

y=ln(x-4)

 

if you let x be any negative such as -1, -2, -5, ...-infinity

 

chose x = -1

 

y = ln (-1 - 4) , y = ln (-5) = undefined

 

so ln (negative) is not ideal

 

if x = 1, y = ln (1-4) = ln (-3) = error

 

if x = 4 then y = ln (4 -4) = ln (0) = - infinity

 

so the domain and range  of y = ln (x - 4)

 

Domain : [4, infinity) this means the domain is closed at 4 and is open to infinity

 

Range: [ - infinity, infinity)

 

Use thi guide line to solve the rest

Answer 2

y = ln(x-4)

 

x–4 > 0 ; GREATER than zero, NOT ≥ 0. Try to find ln(0) on your calculator; it doesn't give you a number.

 

So Domain is x > 4, or in Interval Notation (4,+∞).

 

Range is All Real Numbers, or in Interval Notation (–∞,+∞).

 

It really helps to visualize the problem if you can sketch y = ln(x) and then translate it 4 units to the right.