Question

graph ( 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 \( 2>1 \). Its properties:

• Domain: \( (-\infty, \infty) \)
• Range: \( (0, \infty) \)
• Passes through \( (0, 1) \) (since \( 2^0 = 1 \))
• Increasing (as base \( >1 \))

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

The function \( g(x)=\log_2 x \) is a logarithmic function with base \( 2>1 \). Its properties:

• Domain: \( (0, \infty) \)
• Range: \( (-\infty, \infty) \)
• Passes through \( (1, 0) \) (since \( \log_2 1 = 0 \))
• Increasing (as base \( >1 \))
• Also, \( f(x) \) and \( g(x) \) are inverse functions, so their graphs are symmetric about the line \( y = x \).

Step3: Match with Graphs

We need to find a graph where one curve (exponential) passes through \( (0,1) \), is increasing, and the other (logarithmic) passes through \( (1,0) \), is increasing, and they are symmetric about \( y = x \). Looking at the options, the middle - looking graph (assuming the second graph in the vertical list, but more precisely, the graph where the exponential starts at near \( y = 1 \) when \( x = 0 \) and increases, and the logarithmic starts at near \( x = 1 \), \( y = 0 \) and increases, and they are symmetric over \( y=x \)) should be the correct one. But from the visual (assuming the second graph in the given set where the exponential and logarithmic are increasing, pass through their respective points, and are symmetric) is the correct one. Wait, actually, the correct graph should have \( f(x)=2^x \) (exponential, passes through (0,1), increasing) and \( g(x)=\log_2 x \) (logarithmic, passes through (1,0), increasing) and symmetric about \( y = x \). So the graph that shows these two curves with these properties.

Answer:

The Middle Graph (assuming the second graph in the vertical arrangement of the four graphs, with the exponential and logarithmic curves having the correct properties of passing through (0,1) and (1,0) respectively, being increasing, and symmetric about \( y = x \))