Question

question 14 of 25
which of these statements is true for $f(x) = 3 \cdot (9)^{-x}$?

a. the domain of $f(x)$ is $x > 0$.

b. it is always decreasing.

c. the $y$-intercept is $(0, 3)$.

d. the $y$-intercept is $(0, 1)$.

Explanation:

Step1: Analyze Option A

For an exponential function \( f(x) = a\cdot b^x \), the domain is all real numbers (\( x\in\mathbb{R} \)), not \( x > 0 \). So A is false.

Step2: Analyze Option B

The base \( b = 9>1 \), so the function \( f(x)=3\cdot(9)^x \) is an exponential growth function (always increasing), not decreasing. So B is false.

Step3: Analyze Option C

To find the y - intercept, set \( x = 0 \). Then \( f(0)=3\cdot(9)^0=3\cdot1 = 3 \). So the y - intercept is \( (0,3) \), C is true.

Step4: Analyze Option D

From Step 3, we know the y - intercept is \( (0,3) \), not \( (0,1) \). So D is false.

Answer:

C. The \( y \)-intercept is \( (0, 3) \)