Question

here are graphs of two exponential functions, f and g. if ( f(x) = 100 cdot left( \frac{2}{3}
ight)^x ) and ( g(x) = 100 cdot b^x ), what could be the value of b? a. ( \frac{1}{3} ) b. ( \frac{3}{4} ) c. 1 d. ( \frac{3}{2} )

Explanation:

Step1: Analyze exponential decay behavior

For exponential functions of the form $y = a \cdot r^x$ where $a>0$:

• If $0

From the graph, $g(x)$ decays faster than $f(x)$, so its base $b$ must be smaller than $\frac{2}{3}$.

Step2: Compare options to $\frac{2}{3}$

$\frac{2}{3} \approx 0.667$. Evaluate each option:

• A. $\frac{1}{3} \approx 0.333 < 0.667$
• B. $\frac{3}{4} = 0.75 > 0.667$
• C. $1$: A base of 1 gives a constant function, not decay
• D. $\frac{3}{2} = 1.5 > 1$: This would be exponential growth

Answer:

A. $\frac{1}{3}$