Question

If f(x)=In(x) and g(x) is f(x) translated right one unit and down nine units then reflected over the x axis, what would g(30) be?

A. -5.9
B. -12.1
C. -12.4
D. -20.1​

Answer 1

Translations

`f(x+a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units left}`

`f(x-a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units right}`

`f(x)+a \implies f(x) \: \textsf{translated}\:a\:\textsf{units up}`

`f(x)-a \implies f(x) \: \textsf{translated}\:a\:\textsf{units down}`

`y=-\:f\:(x) \implies f(x) \: \textsf{reflected in the} \: x \textsf{-axis}`

`y=f\:(-\:x) \implies f(x) \: \textsf{reflected in the} \: y \textsf{-axis}`

Parent function:
`f\:(x) = \ln(x)`

Translated right 1 unit:
`f\:(x\:-1) = \ln(x - 1)`

Then translated down 9 units:
`f\:(x\: -1)-9 = \ln(x - 1) - 9`

The reflected over the x-axis:
`-\:[f\:(x\:-1) - 9] = -\ln(x - 1) + 9`

Therefore,
`g(x) = -\ln\:(x\:- 1) + 9`

⇒ g(30) = - ln(30 - 1) + 9

= -3.36729... + 9

= 5.6 (nearest tenth)