Question

Given the parent function f(x)=ln(x)(a) Write a function whose graph has been reflected in the x-axis, is three times as tall, and has been shifted four units to the left.(b) Write a function whose graph has been reflected in the y-axis, has been shifted seven units up, and is half as wide.(c) Calculate the inverse function for your answer to question (b).

Answer 1

Given the function

$f(x)=\ln (x)$

a) When we multiply the parent function by –1, we get a reflection about the x-axis. Then

$g(x)=-\ln (x)$

And three times as tall means 3 units up

$g(x)=-\ln (x)+3$

Shifted four units to the left means shifted 4 units left

$g(x)=-\ln (x+4)+3$

Answer: the new function is: g(x) = - ln(x+4) + 3

b) When we multiply the input by –1, we get a reflection about the y-axis.

$g(x)=\ln (-x)$

Shifted seven units up

$g(x)=\ln (-x)+7$

Half as wide means 1/2 f(x)

$g(x)=\ln (-\frac{x}{2})+7$

Answer: the new function is:

$g(x)=\ln (-\frac{x}{2})+7$

c) We find the inverse of

$g(x)=\ln (-\frac{x}{2})+7$

Then by definition of inverses, g(x) = y

$y=\ln (-\frac{x}{2})+7$

Next, replace all x’s with and all y’s with x

$x=\ln (-\frac{y}{2})+7$

Now, solve for y

$\begin{gathered} x-7=\ln (-\frac{y}{2})+7-7 \\ x-7=\ln (-\frac{y}{2}) \end{gathered}$

Apply properties of logarithms

$\begin{gathered} e^{x-7}=-\frac{y}{2} \\ 2\cdot e^{x-7}=-\frac{y}{2}\cdot2 \\ 2e^{x-7}=-y \\ y=-2e^{x-7} \end{gathered}$

Answer:

$g(x)^{-1}=-2e^{x-7}$