A sports trainer has monthly costs of \( \$ 67.67 \) for phone service and \( \$ 44.42 \) for a website and advertising. In addition, \( \$ 15 \) fee is paid to the \( 9 y m \) for each session in which the trainer works with a client. Part 1 of 4 (a) Write a cost function to represent the cost \( C(x) \) for \( x \) training sessions. \[ C(x)=112.09+15 x \] Part: \( 1 / 4 \) Part 2 of 4 (b) Write a function representing the average cost \( \bar{C}(x) \) for \( x \) sessions. ( \( \bar{C}(x)=\square \)

To find the average cost function \( \bar{C}(x) \), you can divide the total cost function \( C(x) \) by the number of sessions \( x \). The total cost is given as \( C(x) = 112.09 + 15x \). So the average cost function becomes: \[ \bar{C}(x) = \frac{C(x)}{x} = \frac{112.09 + 15x}{x} = \frac{112.09}{x} + 15 \] This equation shows that as the number of sessions \( x \) increases, the average cost per session decreases, except for the fixed part of the cost which will become less significant. Now, let's get into a fun fact. Did you know that training sessions have been quantitatively assessed since the early days of competitive sports? Innovators like John Graham in the early 20th century laid the groundwork for this kind of analysis, which has become a vital ingredient in modern sports training! In real-world applications, knowing how to calculate average costs can significantly affect decision-making for a sports trainer. A trainer might determine whether to take on more clients based on how the average cost per session declines with increased clients. It's a strategic way to maximize profit while keeping quality training accessible.