Put strawberries into a blender and a smoothie comes out; put carrots into a blender and chopped carrots come out. A function is the same: it produces one output for each individual input and the same input cannot produce two different outputs. For example, you cannot put strawberries into a blender and get both a smoothie and chopped carrots. This is what mathematicians mean when they write a function such as f(x) = x + 1. Put strawberries (x) into the function, and you get a smoothie (x + 1).

- Paper
- Pencil
- Calculator

Write the ordered pairs you want to analyze. For example, write, (3,7) and (7,2).

Write the quotient of the difference of the second term of the second pair and the second term of the first pair divided by the difference of the first term of the second pair and the first term of the first pair. Solve using a calculator. For example, (2 - 7)/(7 - 3) = -1.25.

Substitute your answer as the value of the variable m in the equation y = mx + b. For example, write, y = -1.25x + b.

Substitute the first term of the first ordered pair into the same equation in place of the variable x. For example, write, y = (-1.25 x 3) + b.

Substitute the second term of the first ordered pair into the same equation in place of the variable y. For example, write, 7 = (-1.25 x 3) + b.

Simplify your equation by completing the multiplication in the parentheses using a calculator. For example, write, 7 = -3.75 + b.

Simplify your equation again by adding a term to both sides of the equation that will leave the b variable alone on its side of the equation. For example, if you add 3.75 to both sides of the equation, 3.75 and -3.75 on the right side of the equation will cancel, leaving the variable b alone. Write, 7 + 3.75 = -3.75 + 3.75 + b.

Simplify your equation by performing the indicated addition operations. For example, write, 10.75 = b.

Substitute your answer for the variable b in the original equation y = mx + b. For example, write, y = mx + 10.75.

Substitute into the same equation your original value for m. For example, your original value for m was -1.25. Write, y = -1.25x + 10.75. You have calculated a function from the ordered pairs (3,7) and (7,2).

#### Things You'll Need

References

- Purplemath; Functions versus Relations; Elizabeth Stapel; 2011
- West Texas A&M University; Intermediate Algebra Tutorial 13: Introduction to Functions
- University of Colorado, Boulder; 1.5: Functions
- Algebra-Class.com: Linear Functions
- Kennesaw State University; Linear Functions; S. F. Ellermeyer; 2001
- North Seattle Community College; 3.4: The Point-Slope Form of a Line

About the Author

John Woloch writes professionally for various websites. He has published in the Dutch journal "Crux" and writes frequently on oil painting, classical languages and topics involving math and biochemistry. Woloch holds a Master of Arts in English from the University of Chicago, a Master of Arts in classics from Ohio State University and a postbaccalaureate pre-medical degree from Georgetown University.

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