Question

which best describes the range of the function $f(x) = \frac{2}{3}(6)^x$ after it has been reflected over the $x$-axis?
all real numbers
all real numbers greater than 0
all real numbers less than or equal to 0
all real numbers less than 0

Explanation:

Step1: Find range of original function

The original function is \( f(x)=\frac{2}{3}(6)^x \). For an exponential function \( a^x \) (where \( a > 1 \)), the range is \( (0,+\infty) \). So for \( f(x)=\frac{2}{3}(6)^x \), since \( \frac{2}{3}>0 \) and \( 6^x>0 \) for all real \( x \), the range of \( f(x) \) is \( (0,+\infty) \), meaning all real numbers greater than 0.

Step2: Effect of reflection over x - axis

Reflecting a function \( y = f(x) \) over the \( x \) - axis gives the new function \( y=-f(x) \). If the range of \( f(x) \) is \( (0,+\infty) \) (all real numbers greater than 0), then the range of \( -f(x) \) will be all real numbers less than 0. Because if \( y = f(x)>0 \), then \( y=-f(x)<0 \).

Answer:

all real numbers less than 0