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Exercises Functions

Example 13 from Algebra and Trigonometry, 3.1 Functions and Function Notation

Determining Whether a Relationship Is a One-to-One Function

Is the area of a circle a function of its radius? If yes, is the function one-to-one?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
A circle of radius $r$ has a unique area measure given by $A = π r^{2},$ so for any input, $r,$ there is only one output, $A .$ The area is a function of radius $r .$
If the function is one-to-one, the output value, the area, must correspond to a unique input value, the radius. Any area measure $A$ is given by the formula $A = π r^{2} .$ Because areas and radii are positive numbers, there is exactly one solution: $\sqrt{\frac{A}{π}} .$ So the area of a circle is a one-to-one function of the circle’s radius.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

ⓐ Is a balance a function of the bank account number?
ⓑ Is a bank account number a function of the balance?
ⓒ Is a balance a one-to-one function of the bank account number?

How did it go?

PreviousReading Function Values from a Graph Next Applying the Vertical Line Test

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 3.1, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.