Answer:
[tex]g(x)=\frac{1}{2}(3^{-x})[/tex]
Step-by-step explanation:
Given:
The graph of function [tex]f(x)\ and\ g(x)[/tex] are given.
The equation for [tex]f(x)[/tex] is given as:
[tex]f(x)=3^x[/tex]
Now, the graph of [tex]g(x)[/tex] is a reflection of [tex]f(x)[/tex].
The graph of [tex]g(x)[/tex] passes through the point (-1, 1.5) [second quadrant] and crosses the y-axis at (0, 0.5).
As evident from the graph, the functions [tex]f(x)\ and\ g(x)[/tex] are reflections about the y-axis.
We know the transformation rule for reflection about the y-axis as:
[tex]f(x)\to f(-x)\\\\\therefore 3^x\to3^{-x}....(\textrm{Reflection about y-axis})[/tex]
Now, the y-intercept of the function [tex]y=3^{-x}[/tex] is obtained by plugging in [tex]x=0[/tex]. This gives,
[tex]y=3^0=1[/tex]
So, the y-intercept is at (0, 1). But the graph of [tex]g(x)[/tex] crosses the y-axis at (0, 0.5). As we observe, the coordinate rule for the transformation can be written as:
(0, 1) → (0, 0.5)
[tex](x,y)\to (x,\frac{1}{2}y)[/tex]
So, the reflected graph is compressed vertically by a factor of [tex]\frac{1}{2}[/tex].
Therefore, the transformation is given as:
[tex]y\to \frac{1}{2}y\\\\\therefore 3^{-x}\to \frac{1}{2}(3^{-x})[/tex]
Therefore, the equation for [tex]g(x)[/tex] is:
[tex]g(x)=\frac{1}{2}(3^{-x})[/tex]
