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]
