The problem describes a situation in which a new television depreciates by $110 per year and is worth $160 after six years. We are asked to determine a function that models the value of the television as a function of its age $x$ in years.

Step-by-step solution:

1. Linear depreciation model:

   • The value of the television decreases by $110 each year, which suggests a linear depreciation model of the form: $f(x) = -110x + b$ where $f(x)$ represents the value of the television after $x$ years, and $b$ is the initial value of the television (when $x = 0$).

2. Finding the initial value:

   • We are told that after 6 years, the television is worth $160. So, when $x = 6$, $f(6) = 160$. Plugging this into the equation gives: $160 = -110(6) + b$ $160 = -660 + b$ Solving for $b$: $b = 160 + 660 = 820$ Therefore, the initial value of the television was $820.

3. The final function: The function that describes the value of the television after $x$ years is: $f(x) = -110x + 820$

Final answer:

$f(x) = -110x + 820$

Would you like further details or explanations about any step?

Here are some related questions to deepen your understanding:

1. How does linear depreciation work in general terms?
2. What happens to the value of the television after 10 years based on this model?
3. Can you derive the general form of a linear function from two points?
4. How would the model change if the depreciation rate was different?
5. How can we interpret the intercept $b$ in practical terms for this problem?

Tip: Linear functions are great tools to model constant rates of change, like depreciation, because they show a straight-line relationship between the variables.