Question

graphs f(x) = 2^x and g(x) = \log_2 x in the same rectangular coordinate system. choose the correct graph below.

Explanation:

Step1: Analyze \( f(x) = 2^x \)

The function \( f(x)=2^x \) is an exponential function with base \( 2>1 \), so it is increasing, passes through \( (0,1) \) (since \( 2^0 = 1 \)), and has a horizontal asymptote at \( y = 0 \) as \( x
ightarrow-\infty \).

Step2: Analyze \( g(x)=\log_2 x \)

The function \( g(x)=\log_2 x \) is a logarithmic function with base \( 2>1 \), so it is increasing, passes through \( (1,0) \) (since \( \log_2 1 = 0 \)), and has a vertical asymptote at \( x = 0 \) as \( x
ightarrow0^+ \).

Step3: Match the Graphs

We need a graph where one curve is the increasing exponential \( 2^x \) (passing through \( (0,1) \)) and the other is the increasing logarithmic \( \log_2 x \) (passing through \( (1,0) \)). Looking at the options, the third graph (from the top) should have the exponential curve (increasing, through \( (0,1) \)) and the logarithmic curve (increasing, through \( (1,0) \)).

Answer:

The third graph (from the top)