Question

the graph of $f(x) = \frac{1}{2}(2.5)^x$ and its reflection across the x-axis, $g(x)$, are shown. what is the range of $g(x)$? \bigcirc all real numbers \bigcirc all real numbers less than 0 \bigcirc all real numbers greater than 0 \bigcirc all real numbers less than or equal to 0

Explanation:

Step1: Analyze the original function's range

The function \( f(x)=\frac{1}{2}(2.5)^x \) is an exponential function. For an exponential function \( a^x \) where \( a > 1 \), the range of \( f(x)=k\cdot a^x \) (here \( k=\frac{1}{2}>0 \)) is all real numbers greater than \( 0 \), because \( (2.5)^x>0 \) for all real \( x \), and multiplying by \( \frac{1}{2} \) (a positive constant) still keeps the result positive. So the range of \( f(x) \) is \( y>0 \).

Step2: Analyze the reflection across the x - axis

When we reflect a function \( y = f(x) \) across the \( x \) - axis, the new function \( g(x) \) is given by \( g(x)=-f(x) \). So if \( f(x)=\frac{1}{2}(2.5)^x \), then \( g(x)=-\frac{1}{2}(2.5)^x \).

Since the range of \( f(x) \) is \( y > 0 \), when we multiply by \( - 1 \), the inequality sign flips. So the range of \( g(x) \) is \( y<0 \). We can also see from the graph that the graph of \( g(x) \) is below the \( x \) - axis (where \( y = 0 \)), so all the \( y \) - values of \( g(x) \) are less than \( 0 \).

Answer:

all real numbers less than 0