Does the function f of x equals 3 times the quantity 1 over 2 end quantity to the x power represent growth or decay? What is the y-intercept of f(x)?

Question

a Growth; 0 comma one-half
b Growth; (0, 3)
c Decay; 0 comma one-half
d Decay; (0, 3)

Answer 1

To determine whether the function \( f(x) = 3 \left( \frac{1}{2} \right)^x \) represents growth or decay, we need to look at the base of the exponential term \(\left( \frac{1}{2} \right)^x\).

1. Growth or Decay:
   • If the base of the exponential function is greater than 1, the function represents growth.
   • If the base is between 0 and 1 (0 < base < 1), the function represents decay.

In this case, the base is \(\frac{1}{2}\), which is less than 1. Therefore, \( f(x) \) represents decay.

2. Finding the y-intercept: The y-intercept of a function is found by evaluating the function at \( x = 0 \): \[ f(0) = 3 \left( \frac{1}{2} \right)^0 = 3 \cdot 1 = 3 \]

So the y-intercept is \( (0, 3) \).

Considering both the decay nature and the y-intercept, the correct answer is:

d Decay; (0, 3).