3.1 Functions

1. 3.1 Functions Determine whether a relation is a function. Find the domain of functions. Evaluate piecewise-defined and greatest integer functions.

2. Example #1Determining Inputs and Outputs of Functions Describe the set of inputs, the set of outputs, and the rule for the following functions: • The cost of a tank of gasoline depends on the number of gallons purchased. • The volume of a cube depends on the length of its edges. Input: Number of gallons purchased Output: Cost of a gas Rule: Price per gallon times number of gallons Input: Length of edge of cube Output: Volume of cube Rule: V = s3

3. Example #1Determining Inputs and Outputs of Functions • A hydrologist records the depth of a pond in a certain place over a 6 month period in the form of a graph. The graph shows the depth recorded each month. Input: Month Output: Depth Rule: Month/Depth Graph

4. Example #2Determining Whether a Relation is a Function Yes, every input has exactly one output, even though 1 & 3 share the same output it is still a function. Yes, every input has exactly one output, even though 10 repeats as an input twice, the output is the same.

5. Example #3Evaluating a Function • Find the indicated values of • g(1) • g(−3) • g(y+2)

6. Example #4Finding a Difference Quotient Find each output for • First find

7. Example #4Finding a Difference Quotient Find each output for • Next find

8. Example #4Finding a Difference Quotient Find each output for • Finally find

9. Example #5Determining if an Equation Defines a Function Solve each equation for y, graph, and use the vertical line test to determine whether the line is a function. It is a function, a vertical line only crosses once.

10. Example #5Determining if an Equation Defines a Function Solve each equation for y, graph, and use the vertical line test to determine whether the line is a function. Even roots have two possible answers and require the ± symbol. Not a function, a vertical line will cross the line twice.

11. Domains of Functions • The definition of domain is the set of all “input” values for which the function is defined. Defined is the key word here because any values that make the function undefined are excluded from the domain. • There are two situations that will make a function undefined: • Dividing by 0. • Taking an even root (for instance a square root) of a negative number, which produces imaginary solutions. • To find the domain of a function, we must determine what value(s) of x, if any, will either make the denominator become zero OR make the radicand of an even root negative.

12. Example #6Finding Domains of Functions • Find the domain for each function given. If x is in the denominator, we must determine what values of x would make the denominator become zero, because 3/0 is undefined. We do this by setting the denominator equal to zero and solving, remember, this value is excluded from the domain! Therefore the domain of this function is all reals except −4.

13. Example #6Finding Domains of Functions • Find the domain for each function given. First of all, notice that the denominator is NOT factorable because it has a +4 and not a -4. In this case, the only excluded values are imaginary, therefore the domain is the set of all reals.

14. Example #6Finding Domains of Functions • Find the domain for each function given. This time we have a radical, specifically a square root, so we must find the values of w which would make the radicand negative. If the radicand cannot be negative, it must be greater than or equal to zero, because the square root of 0 is 0, and therefore defined. The domain is then all reals less than or equal to 1/3.

15. Example #6Finding Domains of Functions • Find the domain for each function given. This time the root is a cube root and has no restrictions on its domain.

16. Example #7 • Robin cleans small passenger airplane interiors at an airport. At most, she will clean 24 planes per month. Robin pays a license fee of $150 per month to the airport. Her cost per plane (cleaning supplies, commute to airport, and salary of a helper) is approximately $82.50. She charges $229 to clean each plane. Let x represent the number of planes she cleans in a month. • Express Robin’s monthly revenue R as a function of x. Revenue is the gross amount of money made without taking expenses into consideration.

17. Example #7 • Robin cleans small passenger airplane interiors at an airport. At most, she will clean 24 planes per month. Robin pays a license fee of $150 per month to the airport. Her cost per plane (cleaning supplies, commute to airport, and salary of a helper) is approximately $82.50. She charges $229 to clean each plane. Let x represent the number of planes she cleans in a month. • Express the monthly cost C as a function of x.

18. Example #7 • Robin cleans small passenger airplane interiors at an airport. At most, she will clean 24 planes per month. Robin pays a license fee of $150 per month to the airport. Her cost per plane (cleaning supplies, commute to airport, and salary of a helper) is approximately $82.50. She charges $229 to clean each plane. Let x represent the number of planes she cleans in a month. • Find the rule and the domain of the monthly profit function P. Since she can’t clean negative amounts of planes & at most 24 planes, the domain is [0, 24]

19. Example #8Evaluating a Piecewise-Defined Function Given: • h(0) • h(3.5) • h(4) 0 is in the domain of the second function because 0 < 3. 3.5 does not belong to the domain of either function and is therefore undefined. 4 is in the domain of the first function because 4 ≥ 4.

20. Greatest Integer Function • For any integer x, round down to the nearest integer less than or equal to x. OR Sometimes this is called the floor function instead of the greatest integer function.

21. Example #9Evaluating the Greatest Integer Function Let Evaluate: Don’t forget to round down. 6 7 -210 (remember that -210 < -209) -51 3 (the square root of 10 is approximately 3.2) 1 (5/4 is 1.25)