Questions: 3) Graph of y=B(t) a) What is the domain of B(t)? Write it in reduced interval notation. b) What is the range of B(t)? Write it in reduced interval notation. c) Evaluate B(2). d) Pretend that ε>0 is a very small positive number. You don't know its specific value. It is just really small. Explain why this statement is false: When t ∈(2-ε, 2+ε), then B(t) is close to -6.

3) Graph of y=B(t)
a) What is the domain of B(t)? Write it in reduced interval notation.
b) What is the range of B(t)? Write it in reduced interval notation.
c) Evaluate B(2).
d) Pretend that ε>0 is a very small positive number. You don't know its specific value. It is just really small. Explain why this statement is false:

When t ∈(2-ε, 2+ε), then B(t) is close to -6.

Transcript text: 3) Graph of $y=B(t)$ a) What is the domain of $B(t)$ ? Write it in reduced interval notation. b) What is the range of $B(t)$ ? Write it in reduced interval notation. c) Evaluate $B(2)$. d) Pretend that $\varepsilon>0$ is a very small positive number. You don't know its specific value. It is just really small. Explain why this statement is false: When $t \in(2-\varepsilon, 2+\varepsilon)$, then $B(t)$ is close to -6.

Solution

Solution Steps

Step 1: Finding the domain of B(t)

The domain of a function is the set of all possible input values (t-values in this case). The graph of B(t) extends from t = -8 (inclusive, as indicated by the solid dot) to t = -4 (exclusive, as indicated by the open circle). Then it picks up again at t = -2 (exclusive) and continues towards positive infinity. Thus, the domain in interval notation is $[-8, -4) \cup (-2, \infty)$.

Step 2: Finding the range of B(t)

The range of a function is the set of all possible output values (y-values or B(t) values in this case). Observing the graph, the lowest point is approximately -6 and it doesn't reach 7 (open circle at y=7 and y=3 for the separate parts). Combining the separate parts, the range of B(t) is approximately $[-6, 7)$.

Step 3: Evaluating B(2)

To evaluate B(2), we look for the y-value when t = 2. The graph shows a small open circle at t = 2 and y = -2 approximately. Since the circle is open, we look for the value it approaches. This seems to approach -2, so $B(2) \approx -2$.

Final Answer

a) $\boxed{[-8, -4) \cup (-2, \infty)}$ b) $\boxed{[-6, 7)}$ c) $\boxed{B(2) \approx -2}$ d) The statement "When $t \in (2 - \varepsilon, 2 + \varepsilon)$, then $B(t)$ is close to -6" is false because, as observed from the graph, when _t_ is close to 2, _B(t)_ is close to -2, not -6. The value of -6 is attained when t is close to 3.

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