Does the function [tex][tex]$f(x)=4\left(\frac{1}{3}\right)^x$[/tex][/tex] represent growth or decay? What is the [tex][tex]$y$[/tex][/tex]-intercept of [tex][tex]$f(x)$[/tex][/tex]?

A. Growth; [tex][tex]$(0,4)$[/tex][/tex]
B. Growth; [tex][tex]$\left(0, \frac{1}{3}\right)$[/tex][/tex]
C. Decay; [tex][tex]$(0,4)$[/tex][/tex]
D. Decay; [tex][tex]$\left(0, \frac{1}{3}\right)$[/tex][/tex]



Answer :

To determine whether the function [tex]\( f(x) = 4\left(\frac{1}{3}\right)^x \)[/tex] represents growth or decay and to find the [tex]\( y \)[/tex]-intercept, we need to analyze the structure of the function and its components.

1. Identify the Base and Coefficient:
- The given function is [tex]\( f(x) = 4\left(\frac{1}{3}\right)^x \)[/tex].
- The base of the exponential expression is [tex]\( \frac{1}{3} \)[/tex].
- The coefficient is 4.

2. Determine Growth or Decay:
- In an exponential function of the form [tex]\( g(x) = a \cdot b^x \)[/tex], if [tex]\( b \)[/tex] (the base) is between 0 and 1, the function represents exponential decay.
- Here, the base [tex]\( \frac{1}{3} \)[/tex] is between 0 and 1.
- Therefore, the function [tex]\( f(x) = 4\left(\frac{1}{3}\right)^x \)[/tex] represents decay.

3. Find the [tex]\( y \)[/tex]-Intercept:
- The [tex]\( y \)[/tex]-intercept of a function is found by evaluating the function at [tex]\( x = 0 \)[/tex].
- Plugging in [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 4\left(\frac{1}{3}\right)^0 \][/tex]
- Any number raised to the power of 0 is 1, so:
[tex]\[ \left(\frac{1}{3}\right)^0 = 1 \][/tex]
- Thus:
[tex]\[ f(0) = 4 \cdot 1 = 4 \][/tex]

- Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, 4) \)[/tex].

Putting it all together:

- The function [tex]\( f(x) \)[/tex] represents decay.
- The [tex]\( y \)[/tex]-intercept is [tex]\( (0, 4) \)[/tex].

The correct answer is:

Decay; [tex]\( (0, 4) \)[/tex]