Question

select the correct answer.
which function is decreasing and approaches negative infinity as x increases?
a. f(x) = 3(0.6)^x - 1
b. f(x) = -3(6)^x + 2
c. f(x) = -3(0.6)^x + 1
d. f(x) = 3(6)^x - 2

Explanation:

Step1: Analyze exponential term behavior

For exponential functions $a(b)^x$: if $b>1$, $b^x \to \infty$ as $x\to\infty$; if $0

Step2: Check each option for decreasing and limit

• A: $3(0.6)^x -1$: $0.6^x\to0$, so $f(x)\to-1$ (not $-\infty$)
• B: $-3(6)^x +2$: $6^x\to\infty$, so $-3(6)^x\to-\infty$, $f(x)\to-\infty$; derivative $-3\ln6(6)^x<0$ (decreasing)
• C: $-3(0.6)^x +1$: $0.6^x\to0$, so $f(x)\to1$ (not $-\infty$)
• D: $3(6)^x -2$: $6^x\to\infty$, $f(x)\to\infty$ (increasing)

Step3: Confirm correct option

Only B is decreasing and approaches $-\infty$ as $x$ increases.

Answer:

B. f(x) = -3(6)^x + 2