Question

20 POINTS!! Two exponential functions are shown in the table.

Answer 1

Answer:

The correct option is 2.

Step-by-step explanation:

The given functions are

$f(x)=2^x$

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

It can be written as

$g(x)=(2)^{-x}$

When a function is reflected about the y-axis then

$(x,y)\rightarrow (-x,y)$

Since f(-x)=g(x), therefore functions f(x) and g(x) are reflections over the y-axis. Option 1 is incorrect and option 2 is correct.

The general exponential function is

$h(x)=ab^x$

Where, a is initial value and b is growth factor.

The function f(x) is increasing function because b=2>1.

The function g(x) is decreasing function because b=1/2<1.

Therefore option 3 is incorrect.

$f(0)=2^0=1$

$g(0)=(\frac{1}{2})^0=1$

The initial value of both functions is 1.

Therefore option 4 is incorrect.

Answer 2

Answer-

The functions f(x) and g(x) are reflections over the y-axis.

Solution-

While reflecting any point with coordinates (x, y) over y-axis or x=0 line, the point shifts to (-x, y)

$(x,y)\rightarrow (-x,y)$

While reflecting any point with coordinates (x, y) over x-axis or y=0 line, the point shifts to (x, -y)

$(x,y)\rightarrow (x,-y)$

The given functions are,

$f(x)=2^x,\ g(x)=(\frac{1}{2})^x$

Applying the formula for reflection of f(x) over y axis would be,

$\Rightarrow y=2^{-x}$

$\Rightarrow y={(2^{-1})}^{x}$

$\Rightarrow y={(\frac{1}{2})}^{x}$

This is the equation for g(x)

Therefore, f(x) and g(x) are reflections over the y-axis.