multiple choice 2 points the function h(x)=ln−(x + 1)−2 is a transformation of the function f(x)=lnx. which…

Answer:

The graph of $h(x)$ is the graph of $f(x)$ reflected over the y - axis, translated 1 unit to the left and 2 units down.

Explanation:

Step1: Analyze the sign change inside the logarithm

For $y = f(x)=\ln x$ and $y = h(x)=\ln[-(x + 1)]-2$, the negative sign in front of $(x + 1)$ inside the logarithm function $\ln[-(x + 1)]$ reflects the graph of $y = \ln x$ over the y - axis. When we have $y=\ln(-x)$ compared to $y = \ln x$, it is a y - axis reflection.

Step2: Analyze the horizontal translation

The $x+1$ inside the logarithm function indicates a horizontal translation. The general rule for a horizontal translation of $y = f(x)$ to $y=f(x + c)$ is a shift of $|c|$ units to the left when $c>0$. Here $c = 1$, so there is a 1 - unit left - ward translation.

Step3: Analyze the vertical translation

The subtraction of 2 outside the logarithm function, i.e., $y=\ln[-(x + 1)]-2$, indicates a vertical translation. The general rule for a vertical translation of $y = f(x)$ to $y=f(x)-k$ (where $k>0$) is a shift of $k$ units down. Here $k = 2$, so there is a 2 - unit downward translation.