Question

1. a recent newspaper gave temperature data for various days of the week in table format. in which of the tables below is the reported temperature a one - to - one function of the day of the week? (1)

x	y
tue	68
wed	65
thu	74

(2)

x	y
tue	72
wed	68
thu	72

(3)

x	y
tue	52
mon	81
tue	76

(4)

x	y
tue	58
mon	85
tue	85

1. physics students drop a basketball from 8 feet above the ground and measure its height as a function of time until it stops bouncing. (a) sketch a graph of what you believe this function would generally look like. (dont worry about scale on the x - axis.) (b) is the height of the ball a one - to - one function of time? explain your answer.
2. explain why any linear function whose slope is not zero would be a one - to - one function.
3. explain why g(x)=|x| is not a one - to - one function.

Explanation:

Step1: Recall one - to - one function definition

A function \(y = f(x)\) is one - to - one if for every \(y\) value there is exactly one \(x\) value. That is, if \(f(x_1)=f(x_2)\), then \(x_1 = x_2\).

Step2: Analyze each table in question 5

• Table (1): Each day of the week (input \(x\)) has a unique temperature (output \(y\)). So it represents a one - to - one function.
• Table (2): Tuesday and Thursday have the same temperature (\(y = 72\)). So it is not a one - to - one function.
• Table (3): Monday has two different temperatures (\(y=58\) and \(y = 81\)). So it is not a one - to - one function.
• Table (4): Monday has two different temperatures (\(y = 56\) and \(y=85\)), and Tuesday has two different temperatures (\(y = 58\) and \(y = 85\)). So it is not a one - to - one function.

Step3: Analyze question 6(b)

The height of the bouncing ball is not a one - to - one function of time. As the ball bounces, it reaches the same height at different times (on the way up and on the way down). For example, when the ball is going up and then coming down during a bounce, it has the same height at two different moments in time.

Step4: Analyze question 7

Let \(y=mx + b\) be a linear function with \(m
eq0\). Suppose \(y_1=mx_1 + b\) and \(y_2=mx_2 + b\). If \(y_1=y_2\), then \(mx_1 + b=mx_2 + b\). Subtracting \(b\) from both sides gives \(mx_1=mx_2\). Since \(m
eq0\), dividing both sides by \(m\) gives \(x_1=x_2\). So any non - zero slope linear function is one - to - one.

Step5: Analyze question 8

For \(g(x)=|x|\), consider \(x = 1\) and \(x=- 1\). \(g(1)=|1| = 1\) and \(g(-1)=|-1| = 1\). Since \(g(1)=g(-1)\) but \(1
eq - 1\), \(g(x)=|x|\) is not a one - to - one function.

Answer:

• Question 5: Table (1) represents a one - to - one function.
• Question 6(b): No, because the ball reaches the same height at different times.
• Question 7: For \(y=mx + b\) with \(m

eq0\), if \(mx_1 + b=mx_2 + b\), then \(x_1=x_2\).

• Question 8: Because \(g(1)=g(-1)\) but \(1

eq - 1\).