Lowell runs 1.2 miles daily. His goal is to run 120 miles, and he will end his daily runs when he reaches that goal. He uses the equation y=−1.2x+120

Question

to keep track of the distance he still needs to run over time. In this function, x represents the number of days he runs, and y represents the number of miles he must still run to achieve his goal. Complete the inequality to determine the domain of the function given the real-world constraints of the problem.

A function can be represented by the equation g(z)=−z+0.99
. Complete the inequality to determine the range of this function.(1 point)

Use the table to answer the question.

x y
−2
4
0 0
2 −4

4 −8
What is the rate of change of this function from −2
to 2?(1 point)\
\

Jakob draws the graph of a linear function. If the graph includes the points (−4,−2)
and (0,0)
, what is the rate of change of Jakob’s function expressed as a decimal?(1 point)
The rate of change of Jakob’s function is
.

Answer 1

Let's address each part of your question step by step:

1. Domain of Lowell's function:

   • Since Lowell runs 1.2 miles daily until he reaches his goal of 120 miles, we first find out how many days it will take him to reach this goal. The number of days \( x \) can be calculated as: \[ x = \frac{\text{total miles}}{\text{miles per day}} = \frac{120}{1.2} = 100 \text{ days} \]
   • Given that \( x \) cannot be negative, the domain of the function, in terms of days, is given by: \[ 0 \leq x \leq 100 \]
   • Therefore, the inequality representing the domain is: \[ 0 \leq x \leq 100 \]

2. Range of the function \( g(z) = -z + 0.99 \):

   • Since this is a linear function with a negative slope, it decreases as \( z \) increases. The maximum value occurs when \( z \) is at its minimum, and the minimum value occurs when \( z \) is at its maximum.
   • Without specified bounds on \( z \), let's assume the reasonable bounds to determine the practical range of the function. However, if we want to determine the range of this function based on \( z \)'s values:
   • If \( z \) approaches positive infinity, \( g(z) \) approaches negative infinity: \[ g(z) \rightarrow -\infty \]
   • If \( z \) approaches negative infinity, \( g(z) \) approaches positive infinity: \[ g(z) \rightarrow +\infty \]
   • Thus, the range is: \[ (-\infty, 0.99] \]

3. Rate of change from the table (from \( x = -2 \) to \( x = 2 \)):

   • Using the points from the table provided:
     • At \( x = -2 \), \( y = 4 \)
     • At \( x = 2 \), \( y = -4 \)
   • The rate of change can be calculated using the formula: \[ \text{Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]
   • Substituting the values: \[ \text{Rate of Change} = \frac{-4 - 4}{2 - (-2)} = \frac{-8}{4} = -2 \]
   • Therefore, the rate of change from \( -2 \) to \( 2 \) is \(-2\).

4. Rate of change of Jakob’s function from points (−4,−2) to (0,0):

   • Now, using the points given:
     • At \( x = -4 \), \( y = -2 \)
     • At \( x = 0 \), \( y = 0 \)
   • Again, use the formula for rate of change: \[ \text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \]
   • Substituting the values: \[ \text{Rate of Change} = \frac{0 - (-2)}{0 - (-4)} = \frac{2}{4} = 0.5 \]
   • Therefore, the rate of change of Jakob’s function is \( 0.5 \).

In summary:

• Domain of Lowell's function: \( 0 \leq x \leq 100 \)
• Range of \( g(z) \): \( (-\infty, 0.99] \)
• Rate of change from -2 to 2: -2
• Rate of change of Jakob’s function: 0.5