Question

behind you, so lets start fresh today.
functions f (solid) and g (dashed) are graphed.
$f$ is defined as $f(x) = -2\log_2(x + 1) + 3$.
what is the equation of $g$?
choose 1 answer:
\\(\boldsymbol{\text{a}}\\) $g(x) = -2\log_2(x + 1) + 3$
\\(\boldsymbol{\text{b}}\\) $g(x) = 2\log_2(x + 1) - 3$
\\(\boldsymbol{\text{c}}\\) $g(x) = -2\log_2(1 - x) + 3$
\\(\boldsymbol{\text{d}}\\) $g(x) = 2\log_2(1 - x) - 3$

Explanation:

Step1: Identify domain of f

$f(x) = -2\log_2(x + 1) + 3$ has domain $x + 1 > 0 \Rightarrow x > -1$ (right of $x=-1$).

Step2: Observe domain of g

Graph of g is left of $x=1$, so domain $1 - x > 0 \Rightarrow x < 1$ (replace $x$ with $-x$ in log argument: $x+1 \to -x+1=1-x$).

Step3: Verify equation of g

Check if $g(x) = -2\log_2(1 - x) + 3$ matches the graph: it has domain $x < 1$ and matches the shape of f reflected over y-axis, which aligns with the red curve. Other options don't fit the domain or shape.

Answer:

C. $g(x) = -2\log_2(1 - x) + 3$