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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 $t = 0$, Marty is at position $d = 2$.

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. $\left \{(0, 1), (1, 4), (2, 7), (3, 6) \right \}$

B. $\left \{(-2,2), (-1, 1), (0, 0), (1, 3), (2, 4) \right \}$

C. $\left \{(3, 2), (3, 4), (5, 6), (7, 8) \right \}$

D. $\left \{(2, 3), (3, 2), (1, 4), (4, 1) \right \}$

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

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

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

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. $d$ is a function of $t$

B. $t$ is a function of $d$

C. both

D. neither

## Problem 3 – Function Notation

If $f$ is a function of $x$ this can be written as $f(x)$.

For example, $f(x) = x^2$. So $f(3) = 9$.

To use the function ability of your graphing calculator, press $Y=$ and enter $x^2 - 2x + 3$.

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

Press $2^{nd}$ [ENTER] to recall the last entry.

9. For $f(x) = x^2 - 2x + 3$, find $f(4)$ using the graphing calculator, then by substitution showing your work below.

10. For $f(x) = 3x^2 + 5x + 3$, find $f(2)$ 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 $f(x)$ that gives an output of $8.5$?

12. What is the unknown function?

Now select option 2.

13. What is the input for the function $f(x)$ that gives an output of $6$?

14. What is the unknown function?

Now select option 3.

15. What is the input for the function $f(x)$ that gives an output of $83$?

16. What is the unknown function?

Feb 22, 2012

Oct 31, 2014