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_{5}(x + 3) )
b ( g(x)=log_{4}x - 3 )
c ( g(x)=log(x - 3)+4 )
d ( g(x)=3log x + 4 )

Explanation:

Question 12

Step1: Recall the vertical asymptote of \( f(x)=\ln x \)

The parent function \( f(x)=\ln x \) has a vertical asymptote at \( x = 0 \) (since the domain of \( \ln x \) is \( x>0 \), so as \( x \to 0^+ \), \( \ln x \to -\infty \)).

Step2: Analyze vertical asymptote transformations

For a logarithmic function of the form \( \ln(x - h)+k \), the vertical asymptote is at \( x = h \). We need the vertical asymptote to be \( x = 5 \), so we need \( h = 5 \) in the transformed function \( \ln(x - h)+k \).

Step3: Analyze each option

• Option A: \( g(x)=5\ln x - 2 \). The vertical asymptote is still \( x = 0 \) (since there's no horizontal shift), so this is incorrect.
• Option B: \( g(x)=\ln(x + 5)-1 \). The vertical asymptote is \( x=- 5 \) (since \( x+5 = 0\Rightarrow x=-5 \)), so this is incorrect.
• Option C: \( g(x)=\ln(x + 2)+5 \). The vertical asymptote is \( x=-2 \) (since \( x + 2=0\Rightarrow x=-2 \)), so this is incorrect.
• Option D: \( g(x)=\ln(x - 5)+3 \). The vertical asymptote is \( x = 5 \) (since \( x-5=0\Rightarrow x = 5 \)), so this is correct.

Step1: Recall transformation rules for logarithmic functions

• A horizontal translation 3 units left of a function \( y = \log_bx \) is given by \( y=\log_b(x + 3) \) (because for horizontal shifts, \( x\to x + h \) shifts left by \( h \) units when \( h>0 \)).
• A vertical stretch by a factor of \( a \) (where \( a>0 \)) of a function \( y=\log_bx \) is given by \( y = a\log_bx \).

Step2: Analyze each option

• Option A: \( g(x)=4\log_5(x + 3) \). Here, the \( (x + 3) \) represents a horizontal translation 3 units left, and the coefficient 4 represents a vertical stretch by a factor of 4. This matches the required transformations.
• Option B: \( g(x)=\log_4x-3 \). This is a vertical translation 3 units down, with no horizontal shift or vertical stretch as required. So this is incorrect.
• Option C: \( g(x)=\log(x - 3)+4 \). This is a horizontal translation 3 units right and a vertical translation 4 units up, not the required transformations. So this is incorrect.
• Option D: \( g(x)=3\log x + 4 \). This is a vertical stretch by a factor of 3 and a vertical translation 4 units up, with no horizontal shift as required. So this is incorrect.

Answer:

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

Question 13