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# 2.1: Back in Time?

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 1, Lesson 5.

## Definition

A function is a relation in which each input is paired with exactly one output.

For every value that goes into a function, the function outputs one unique result.

## Problem 1 – Graphical

At time t=0\begin{align*}t = 0\end{align*}, Marty is at position d=2\begin{align*}d = 2\end{align*}.

1. Can the graph to the right describe Marty’s position as a function of time? Explain.

2. What would have to happen for this graph to occur?

3. Redraw the dashed lines to make the graph a function.

## Problem 2 – Set of Ordered Pairs

The first element of each ordered pair is the input value.

4. Which sets below describe a function? Explain why.

A. {(0,1),(1,4),(2,7),(3,6)}\begin{align*}\left \{(0, 1), (1, 4), (2, 7), (3, 6) \right \}\end{align*}

B. {(2,2),(1,1),(0,0),(1,3),(2,4)}\begin{align*}\left \{(-2,2), (-1, 1), (0, 0), (1, 3), (2, 4) \right \}\end{align*}

C. {(3,2),(3,4),(5,6),(7,8)}\begin{align*}\left \{(3, 2), (3, 4), (5, 6), (7, 8) \right \}\end{align*}

D. {(2,3),(3,2),(1,4),(4,1)}\begin{align*}\left \{(2, 3), (3, 2), (1, 4), (4, 1) \right \}\end{align*}

Marty flies to Mars, where the acceleration of gravity is 0.375\begin{align*}0.375\end{align*} of what it is on Earth. So with a=12ft/s2\begin{align*}a = 12ft/s^2\end{align*}, use the distance formula d=12at2\begin{align*}d = \frac{1}{2}at^2\end{align*} to compute the output when given the input.

5. Use the formula to compute d\begin{align*}d\end{align*}. Give the set or ordered pairs (t,d)\begin{align*}(t, d)\end{align*} when the input t\begin{align*}t\end{align*} is the set {0,1,2,6}\begin{align*}\left \{0, 1, 2, 6 \right \}\end{align*}.

6. Use the formula to compute t\begin{align*}t\end{align*}. Give the set of ordered pairs (d,t)\begin{align*}(d, t)\end{align*} if the input is d\begin{align*}d\end{align*}. The input set for d\begin{align*}d\end{align*} is {0,23,6}\begin{align*}\left \{0, \frac{2}{3}, 6 \right \}\end{align*}.

7. Which of the two solutions sets from Questions 5 and 6 is a function? Why?

8. From solutions sets above, which is true?

A. \begin{align*}d\end{align*} is a function of \begin{align*}t\end{align*}

B. \begin{align*}t\end{align*} is a function of \begin{align*}d\end{align*}

C. both

D. neither

## Problem 3 – Function Notation

If \begin{align*}f\end{align*} is a function of \begin{align*}x\end{align*} this can be written as \begin{align*}f(x)\end{align*}.

For example, \begin{align*}f(x) = x^2\end{align*}. So \begin{align*}f(3) = 9\end{align*}.

To use the function ability of your graphing calculator, press \begin{align*}Y=\end{align*} and enter \begin{align*}x^2 - 2x + 3\end{align*}.

To enter different values for \begin{align*}x\end{align*} and observe what \begin{align*}f(x)\end{align*} equals, press VARS, arrow right to the Y-VARS menu, select Function and then choose \begin{align*}Y1\end{align*}. Then enter (#), replacing # with the \begin{align*}x-\end{align*}value.

Press \begin{align*}2^{nd}\end{align*} [ENTER] to recall the last entry.

9. For \begin{align*}f(x) = x^2 - 2x + 3\end{align*}, find \begin{align*}f(4)\end{align*} using the graphing calculator, then by substitution showing your work below.

10. For \begin{align*}f(x) = 3x^2 + 5x + 3\end{align*}, find \begin{align*}f(2)\end{align*} using the graphing calculator, then by substitution showing your work below.

## Problem 4 – Function Machine

Run the program MACHINE and select option 1. The program will return an output for the input entered.

11. What is the input for the function \begin{align*}f(x)\end{align*} that gives an output of \begin{align*}8.5\end{align*}?

12. What is the unknown function?

Now select option 2.

13. What is the input for the function \begin{align*}f(x)\end{align*} that gives an output of \begin{align*}6\end{align*}?

14. What is the unknown function?

Now select option 3.

15. What is the input for the function \begin{align*}f(x)\end{align*} that gives an output of \begin{align*}83\end{align*}?

16. What is the unknown function?

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