Question

the graph shows an exponential function, ( f(x) ), and a cubic function. for which interval is ( f(x) ) increasing faster than ( g(x) )? options: ( -2 leq x leq -1 ), ( -3 leq x leq -2 ), ( 2 leq x leq 3 ), ( 0 leq x leq 1 ) (graph included)

Explanation:

Step1: Understand the concept

To determine when \( f(x) \) (exponential) is increasing faster than \( g(x) \) (cubic), we look at the slope (rate of change) of their graphs. A steeper slope means a faster increase.

Step2: Analyze each interval

• For \( -2 \leq x \leq -1 \) and \( -3 \leq x \leq -2 \): \( g(x) \) (cubic) might be increasing or \( f(x) \) could be less steep.
• For \( 0 \leq x \leq 1 \): The exponential might not be steeper than cubic yet.
• For \( 2 \leq x \leq 3 \): Exponential functions grow faster than polynomials (like cubic) for large \( x \), so the slope of \( f(x) \) is steeper here, meaning it's increasing faster.

Answer:

\( 2 \leq x \leq 3 \)