Question

lc algebra 1b - standard (15261)
exponential decay functions
which are true of the function $f(x) = 49\left(\frac{1}{7}\
ight)^x$? choose three correct answers.
as $x$ increases by 1, each $y$-value is one-seventh of the previous $y$-value.
the domain is the set of all real numbers.
the domain is $x > 0$
the range is the set of all real numbers.
the range is $y > 0$

Explanation:

Step1: Analyze domain of $f(x)$

For exponential functions $a(b)^x$, $x$ can be any real number, so domain is all real numbers.

Step2: Analyze range of $f(x)$

$(\frac{1}{7})^x > 0$ for all real $x$, so $49(\frac{1}{7})^x > 0$. Range is $y>0$.

Step3: Analyze rate of change

When $x$ increases by 1:
$f(x+1)=49(\frac{1}{7})^{x+1}=49(\frac{1}{7})^x \cdot \frac{1}{7}=f(x) \cdot \frac{1}{7}$
So each $y$-value is 1/7 of the previous.

Answer:

1. As $x$ increases by 1, each $y$-value is one-seventh of the previous $y$-value.
2. The domain is the set of all real numbers.
3. The range is $y > 0$