Your first impression of functions might intimidate you, but they can be simple. Stay with us as we simplify them for you. Functions are defined by an algebraic expression and have numerical inputs and outputs. Being able to use function notations, evaluating functions for inputs in their domains, and interpreting statements that use function notation in terms of a context is a high emphasis topic on the TASC Test Assessing Secondary Completion™ Mathematics subtest.
Functions can be described in various ways:
- A graph (the trace of a seismograph)
- A verbal rule, “I’ll give you a state, you give me the capital city.”
- An algebraic expression, f(x) = a + bx
- A recursive rule
Functions are represented by a function rule. The function rule explains what's done to the input value to make the output value. For instance, a function rule can be, "Multiply the input value by 2 and then subtract three from it." We could write this function rule as f(x) = 2x – 3.
For an equation to be a function, x is the independent variable and the y value (or f(x) value) is the dependent variable. This makes sense, since the output value depends on the input value.
Interpreting Statements that use Function Notation
The following examples will help you become more proficient at interpreting function notations:
- You choose to rent a car. The car rental company charges a flat rate of $20 plus $0.22 per mile driven. Which function rule applies to this scenario?
- f(m) = 20 + 0.22m
- f(m) = 20m + 0.22
- f(m) = 0.2 + 22m
- f(m) = 22 + 0.20m
Answer: f(m) = 20 + 0.22m. We want the output, f(m), to equal the total cost of renting a car based on the number of miles driven, m. We have an initial constant of $20 and then a slope of $0.22, multiplied by the miles driven, our input. Since the two values are added together, the function we're looking for will be in the form: total cost = 20 + 0.22(miles driven). The only one that takes this form is (a).
- GoLean vitamins are sold by mail order only. They cost $19.99 per bottle. A single $4 shipping and handling fee is added to all orders. Which function rule applies to this scenario?
- f(b) = 19.99 + 4b
- f(b) = 4b + 19.99b
- f(b) = 19.99b – 4
- f(b) = 4 + 19.99b
Answer: f(b) = 4 + 19.99b. The output, f(b) should equal the total cost of ordering b bottles of GoLean vitamins. Since the $4 shipping and handling fee doesn't depend on b, it's a constant. Ordering one bottle is $19.99, but b bottles will cost $19.99b. Since we're adding the two together, the only answer that makes sense is (d).