How do you find the domain and range of #g(x)=ln(x-4)#?

Question

Isaiah Anthony · Accepted Answer

Domain : # x>4# , in interval notation : #(4, oo)#
Range: # g(x) in RR# , in interval notation : #(-oo, oo)# #g(x) = ln(x-4) ; (x-4) >0 or x >4# Domain : # x>4# , in interval notation : #(4, oo)# Range: Output may be any real number. Range: # g(x) in RR# , in interval notation : #(-oo, oo)# graph{ln(x-4) [-20, 20, -10, 10]} [Ans]

Savannah Anderson · Answer

undefined The domain of g(x) = ln(x-4) is (4, ∞), and the range is all real numbers.

Isaiah Anthony · Answer

undefined To find the domain and range of $ g(x) = \ln(x-4) $, follow these steps:

Domain:
1. Determine the values that $ x $ can take in the function.
2. Since the natural logarithm function is defined only for positive real numbers, the expression inside the logarithm, $ x - 4 $, must be greater than 0.
3. Set $ x - 4 > 0 $ and solve for $ x $ to find the domain.
$$ x - 4 > 0 $$
$$ x > 4 $$
4. Therefore, the domain of $ g(x) = \ln(x-4) $ is $ x > 4 $.

Range:
1. Consider the behavior of the natural logarithm function.
2. The natural logarithm function approaches negative infinity as $ x $ approaches 0 from the right, and it approaches positive infinity as $ x $ approaches positive infinity.
3. Since the input of the natural logarithm function is $ x - 4 $, as $ x $ increases, $ x - 4 $ also increases.
4. Therefore, as $ x $ increases without bound, $ \ln(x-4) $ increases without bound.
5. Likewise, as $ x $ approaches 4 from the right (since 4 is the lower bound of the domain), $ x - 4 $ approaches 0 from the right, causing $ \ln(x-4) $ to approach negative infinity.
6. Thus, the range of $ g(x) = \ln(x-4) $ is all real numbers.