Domains, transformations and inverse relationships.
Example 1
Find the real domain of f(x)=ln(4x−28).
y
681012-202 x
f(x)=ln(4x−28)Show worked answer
Answer: x>7
- The logarithm argument must be strictly positive.
- 4x−28>0 gives x>7.
Example 2
For f(x)=ln(x−3)+6, find the x−coordinate of its vertical asymptote.
y
246802.557.5 x
f(x)=ln(x−3)+6Show worked answer
Answer: 3
- The argument tends to zero from above at x=3.
- ln(x−3) tends to−∞, so the vertical asymptote is x=3.
Example 3
f(x)=ex−4+4. Find f−1(5).
y
234560510 x
f(x)=ex−4+4Show worked answer
Answer: 4
- Set ex−4+4=5.
- Then ex−4=1, so x−4=0.
- The inverse value is 4.