Question

the graph below could be the graph of which exponential function? a. $f(x) = 2 \cdot (5)^x$ b. $f(x) = 2 \cdot (0.7)^x$ c. $f(x) = 2^x$ d. $f(x) = 2 \cdot (1.4)^x$

Explanation:

Step1: Analyze the graph's trend

The graph is a decreasing exponential curve, so the base of the exponential function should be between 0 and 1 (since for \( f(x)=a\cdot b^x \), if \( 0 < b < 1 \), it's a decreasing function; if \( b>1 \), it's increasing).

Step2: Evaluate each option

• Option A: \( F(x) = 2\cdot(5)^x \), base \( 5>1 \), increasing function, does not match.
• Option B: \( F(x) = 2\cdot(0.7)^x \), base \( 0.7 \) (between 0 and 1), decreasing function, matches the graph's trend.
• Option C: \( F(x) = 2^x \), base \( 2>1 \), increasing function, does not match.
• Option D: \( F(x) = 2\cdot(1.4)^x \), base \( 1.4>1 \), increasing function, does not match.

Answer:

B. \( F(x) = 2\cdot(0.7)^x \)