Remember we can use any variable to name the function; the notation h(a) us that h depends on a. The value “a” must be put into the function “h” to get a result. Be careful - the parentheses indicate that age is input into the function (Note: do not confuse these parentheses with multiplication!). Function Notation The notation output = f(input) defines a function named f. This would be read “output is f of input” Example 2 A function N = f(y) gives the number of police officers, N, in a town in year y. What does f(2005) = 300 tell us? When we read f(2005) = 300, we see the quantity is 2005, which is a value for the input quantity of the function, the year (y). The output value is 300, the number of police officers (N), a value for the output quantity. Remember N=f(y). So this tells us that in the year 2005 there were 300 police officers in the town. Tables as Functions Functions can be represented in many ways: Words (as we did in the last few examples), tables of values, graphs, or formulas. Represented as a table, we are presented with a list of input and output values. This table represents the age of children in years and their corresponding heights. While some tables show all the information we know about a function, this particular table represents just some of the data available for height and ages of children. a, age in years 5 5 6 7 8 9 10 (output) h, height inches 40 42 44 47 50 52 54 Example 3 Which of these tables define a function (if any)? The and second tables define functions. In both, each input corresponds to exactly one output. The third table does not define a function since the input value of 5 corresponds with two different output values. Solving and Evaluating Functions: When we work with functions, are two typical things we do: evaluate and solve. Evaluating a function is what we do when we know an input, and use the function to determine the
a) Since we cannot take the square root of a negative number, we need the inside of the square root to be non-negative. when . The domain of f(x) is . b) We cannot divide by zero, so we need the denominator to be non-zero. when x = 2, so we must exclude 2 from the domain. The domain of g(x) is . 1.1 Exercises The amount of garbage, G, produced by a city with population p is given by . G is measured in tons per week, and p is measured in thousands of people. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function f. Explain the meaning of the statement . The number of yards of dirt, D, needed to cover a garden with area a square feet is given by . A garden with area 5000 ft2 requires 50 cubic yards of dirt. Express this information in terms of the function g. Explain the meaning of the statement . Select all of the following graphs which represent y as a function of x. a b c d e f Select all of the following graphs which represent y as a function of x. a b c d e f Select all of the following tables which represent y as a function of x. a. b. c. Select all of the following tables which represent y as a function of x. a. b. c.
Example 2 Using the graphs below, evaluate . g(x) f(x) To evaluate , we again start with the inside evaluation. We evaluate using the graph of the g(x) function, finding the input of 1 on the horizontal axis and finding the output value of the graph at that input. Here, . Using this value as the input to the f function, . We can then evaluate this by looking to the graph of the f(x) function, finding the input of 3 on the horizontal axis, and reading the output value of the graph at this input. Here, , so. Composition Formulas When evaluating a composition of functions where we have either created or been given formulas, the concept of working from the inside out remains the same. First we evaluate the inside function using the input value provided, then use the resulting output as the input to the outside function. Example 3 Given and , evaluate . Since the inside evaluation iswe start by evaluating the h(x) function at 1: Then , so we evaluate the f(t) function at an input of 5: We are not limited, however, to using a numerical value as the input to the function. We can put anything into the function: a value, a different variable, or even an algebraic expression, provided we use the input expression everywhere we see the input variable. Example 4 Let and , find f(g(x)) and g(f(x)). To find f(g(x)), we start by evaluating the inside, writing out the formula for g(x) We then use the expression as input for the function f. We then evaluate the function f(x) using the formula for g(x) as the input. Since then This gives us the formula for the composition: