Which of the following patterns shows \( f(0), f(1), f(2) \), and \( f(3) \) for the function \( f(x)=1000\left(\frac{1}{2}\right)^{x} \) ? \( 1000,1000.5,1001,1001.5 \) \( 1000,2000,4000,8000 \) \( 1000,999.5,999,998.5 \) \( 1000,500,250,125 \)

To determine how the function \( f(x) = 1000 \left(\frac{1}{2}\right)^{x} \) behaves at specific values, let's compute \( f(0) \), \( f(1) \), \( f(2) \), and \( f(3) \): - \( f(0) = 1000 \left(\frac{1}{2}\right)^{0} = 1000 \) - \( f(1) = 1000 \left(\frac{1}{2}\right)^{1} = 500 \) - \( f(2) = 1000 \left(\frac{1}{2}\right)^{2} = 250 \) - \( f(3) = 1000 \left(\frac{1}{2}\right)^{3} = 125 \) So, the values of \( f(0), f(1), f(2), f(3) \) are \( 1000, 500, 250, 125 \). Therefore, the correct pattern is \( 1000, 500, 250, 125 \). Now, let’s share some exciting insights! To give some context, exponential functions like this one are fascinating because they model many real-life phenomena such as population decay and radioactive decay! In this case, the function represents a quantity that decreases exponentially over time. It's like watching a timer speedily count down as you wait for your cookies to bake! When understanding functions, a common mistake is confusing exponential decay with linear decay. Remember that exponential functions decrease quickly at first and then slow down, making them quite a spectacle in mathematics!