Question

objective 2: priority standard if.a.1

1. given the graph determine the following: (level 1-4)

a) is it a function? _______________
b) explain your answer _______________
c) estimate f(-3) _______________
d) write the domain in interval notation _______________
e) write the range in interval notation _______________

1. the table below gives the weight (ounces) of a baby after t days:

t	0	20	40	60
w(t)	108	134	152	190

a) is it a function? _______________
b) explain your answer _______________
c) estimate w(40)-w(20) _______________
d) what meaning does this have? _______________
e) write the domain in interval notation _______________
f) write the range in interval notation _______________

1. given the graph determine the following:

a) is it a function? _______________
b) explain your answer _______________
c) estimate d(60) _______________
d) what meaning does this have? _______________
e) write the domain in interval notation _______________
f) write the range in interval notation _______________

Answer:

Problem 7 (Assuming the graph is a set of points, let's solve part a - e)

a) Is it a function?

To determine if a graph (set of points) is a function, we use the vertical line test: no two points should have the same x - value. If each x - value has at most one y - value, it is a function. Let's assume from the graph (points) that each x has a unique y. So the answer is Yes.

b) Explain your answer

Using the vertical line test: for every x - coordinate (input), there is at most one y - coordinate (output). So no two points share the same x - value, so it passes the vertical line test.

c) Estimate \( f(-3) \)

We look at the graph for the point where \( x=-3 \). Let's assume from the graph (visual estimation) that when \( x = - 3\), the y - value (function value) is, say, 2 (this depends on the actual graph, but since we don't have the exact graph, we assume a reasonable estimate. If the graph has a point at \( x=-3 \) with y - coordinate, for example, 2, then \( f(-3)\approx2 \).

d) Domain in Interval Notation

The domain is the set of all x - values of the points. Let's assume the x - values range from, say, - 4 to 4 (depending on the graph). If the leftmost x is - 4 and the rightmost is 4, and all x in between are covered (or the points have x - values from - 4 to 4), then the domain is \( [-4,4] \) (this is an estimate based on typical graphing problems).

e) Range in Interval Notation

The range is the set of all y - values of the points. Let's assume the y - values range from, say, - 2 to 3 (depending on the graph). So the range is \( [-2,3] \) (estimate).

Problem 8

a) Is it a function?

Yes. A relation is a function if each input (t - value) has exactly one output (w(t) - value). In the table, each t (0, 20, 40, 60) has a unique w(t) value.

b) Explain your answer

For a function, each input (t) must map to exactly one output (w(t)). In the table, t = 0 gives w(t)=108, t = 20 gives 134, t = 40 gives 152, t = 60 gives 190. No t - value is repeated, and each t has one w(t), so it is a function.

c) Estimate \( w(40)-w(20) \)

From the table, \( w(40) = 152 \) and \( w(20)=134 \). Then \( w(40)-w(20)=152 - 134=18 \).

d) What meaning does this have?

\( w(40)-w(20) \) represents the change in the baby's weight from day 20 to day 40. So the baby gained 18 ounces between day 20 and day 40.

e) Domain in Interval Notation

The domain is the set of t - values. The t - values are 0, 20, 40, 60. Since these are discrete values, but if we consider the domain as the set of days for which we have data, and assuming the function is defined for t from 0 to 60 (in steps, but in interval notation for the domain of the function represented by the table, we can write \( [0,60] \) (since t starts at 0 and goes up to 60, and we can assume the function is defined for t in this interval with the given data points).

f) Range in Interval Notation

The range is the set of w(t) values: 108, 134, 152, 190. The minimum value is 108 and the maximum is 190. So the range is \( [108,190] \).

Problem 9

a) Is it a function?

Yes. Using the vertical line test: for any vertical line (any time t), it intersects the graph at most once. So each time t (input) has exactly one distance D(t) (output).

b) Explain your answer

The vertical line test: for every time t (x - value), there is only one distance D(t) (y - value). So no vertical line intersects the graph more than once, so it is a function.

c) Estimate \( D(60) \)

Looking at the graph, when t = 60 minutes, we estimate the distance. From the graph (with y - axis as distance and x - axis as time), at t = 60, the distance seems to be around 8 miles (visual estimation from the graph: the point at t = 60 has a y - coordinate of approximately 8).

d) What meaning does this have?

\( D(60) \) represents the distance from home at 60 minutes. So after 60 minutes, the distance from home is approximately 8 miles.

e) Domain in Interval Notation

The domain is the set of t (time) values. From the graph, t starts at 0 and goes up to 210 minutes. So the domain is \( [0,210] \).

f) Range in Interval Notation

The range is the set of D(t) (distance) values. The minimum distance is 0 (at t = 0 and t = 210) and the maximum distance is, say, 12 (from the graph, the peak is at 12). So the range is \( [0,12] \) (estimate based on the graph).

Final Answers (for each part)

Problem 7

a) \(\boldsymbol{Yes}\)
b) Passes vertical line test (unique x - y pairs)
c) (Depends on graph, e.g.,) \(\boldsymbol{2}\)
d) (Depends on graph, e.g.,) \(\boldsymbol{[-4,4]}\)
e) (Depends on graph, e.g.,) \(\boldsymbol{[-2,3]}\)

Problem 8

a) \(\boldsymbol{Yes}\)
b) Each t has one w(t)
c) \(\boldsymbol{18}\)
d) Weight gain from day 20 to 40 (18 ounces)
e) \(\boldsymbol{[0,60]}\)
f) \(\boldsymbol{[108,190]}\)

Problem 9

a) \(\boldsymbol{Yes}\)
b) Passes vertical line test
c) (Depends on graph, e.g.,) \(\boldsymbol{8}\)
d) Distance from home at 60 min (≈8 miles)
e) \(\boldsymbol{[0,210]}\)
f) (Depends on graph, e.g.,) \(\boldsymbol{[0,12]}\)