Question

Which function represents g(x), a reflection of f(x) = On a coordinate plane, 2 exponential functions are shown. g (x) decreases in quadrant 2 and approaches y = 0 in quadrant 1. It goes through (negative 1, 1) and crosses the y - axis at (0, 0.5).(3)x across the y-axis? g(x) = 2(3)x g(x) = −One-half(3)x g(x) = One-half(3)−x g(x) = 2(3)−x

Answer 1

Answer:

$g(x)=\frac{1}{2}(3^{-x})$

Step-by-step explanation:

Given:

The graph of function $f(x)\ and\ g(x)$ are given.

The equation for $f(x)$ is given as:

$f(x)=3^x$

Now, the graph of $g(x)$ is a reflection of $f(x)$.

The graph of $g(x)$ 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 $f(x)\ and\ g(x)$ are reflections about the y-axis.

We know the transformation rule for reflection about the y-axis as:

$f(x)\to f(-x)\\\\\therefore 3^x\to3^{-x}....(\textrm{Reflection about y-axis})$

Now, the y-intercept of the function $y=3^{-x}$ is obtained by plugging in $x=0$. This gives,

$y=3^0=1$

So, the y-intercept is at (0, 1). But the graph of $g(x)$ 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)

$(x,y)\to (x,\frac{1}{2}y)$

So, the reflected graph is compressed vertically by a factor of $\frac{1}{2}$.

Therefore, the transformation is given as:

$y\to \frac{1}{2}y\\\\\therefore 3^{-x}\to \frac{1}{2}(3^{-x})$

Therefore, the equation for $g(x)$ is:

$g(x)=\frac{1}{2}(3^{-x})$

Answer 2

Answer:

g(x)=1/2(3^-x)