Question

1. the function ( f(x)=ln x ) is transformed so that the transformed function has an asymptote of ( x = 5 ). which function could be the transformed function?

a ( g(x)=5ln x - 2 )
b ( g(x)=ln(x + 5)-1 )
c ( g(x)=ln(x + 2)+5 )
d ( g(x)=ln(x - 5)+3 )

1. which function is a translation 3 units left and a vertical stretch by a factor of 4 of its parent function?

a ( g(x)=4log_6(x + 3) )
b ( g(x)=log_4 x - 3 )
c ( g(x)=log(x - 3)+4 )
d ( g(x)=3log x + 4 )

Explanation:

For Question 12:

Step1: Recall asymptote of $\ln x$

The parent function $f(x)=\ln x$ has a vertical asymptote at $x=0$.

Step2: Horizontal shift rule

A horizontal shift of $h$ units right gives $\ln(x-h)$, with asymptote $x=h$.

Step3: Match target asymptote $x=5$

We need $h=5$, so the function is $\ln(x-5)+k$ (vertical shifts don't affect asymptotes).

Step1: Vertical stretch rule

A vertical stretch by factor $a$ multiplies the parent function by $a$: $a\cdot f(x)$.

Step2: Left translation rule

A shift 3 units left replaces $x$ with $x+3$: $f(x+3)$.

Step3: Combine transformations

For parent logarithmic function $f(x)=\log_b x$, the transformed function is $4\log_b(x+3)$.

Answer:

D. $g(x) = \ln(x - 5) + 3$

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For Question 13: