10. Functions One quantity depends on another quantity # of shirts -> revenue Age -> Height Time of day -> Temperature

Function – Definition Formal definition: A function f is a rule that assigns to each element x in a set A exactly one element, f (x), in a set B. ( f (x) is read as “ f of x ”) 4 Representations of a function: Verbally – a description in words Numerically – a table of values Algebraically – a formula Visually – a graph

Verbal & Numeric Verbal: Function f(x) divides x by 3 and then adds 4. Table describes the values in A and their assignments in B. Numeric: x -2 5 9 -4 1 f(x)

Algebraic & Graphic Algebraic: Graphic: Plot points (x, f(x)) in the two dimensional plane f(x) x -2 5 9 -4 1 f(x) 42 21 x 2 4 6 8 -21

Example: Evaluate the function at the indicated values.

Piecewise Functions Evaluate the function at the indicated values.

Difference Quotient Find f(a), f(a+h), and the difference quotient for the given function. Difference quotient:

Domain Set A = input => x = independent variable Recall definition: A function f is a rule that assigns to each element x in a set A exactly one element, f(x), in a set B. Set A = input => x = independent variable Set B = output => y = f(x) = dependent variable Domain: Two restrictions on the domain for any function:

Example 1 Find the domain for the following function. D:

Example 2 Find the domain for the following function. D:

11. Graphs of Functions Functions in one variable can be represented by a graph. Each ordered pair (x, f(x)) that makes the equation true is a point on the graph. Graph function by plotting points and then connecting the points with smooth curves.

Example f(x) Create a table of points: 6 x -3 -2 -1 1 x -1 1 -6

Basic Functions - Linear f(x) = mx +b f(x) f(x) x x

Basic Functions - Power f(x) = f(x) f(x) x x

Basic Functions - Root f(x) = f(x) f(x) x x

Basic Functions - Reciprocal f(x) = f(x) f(x) x x

Basic Functions – Absolute Value f(x) = f(x) x

Vertical Line Test y= Determine if an equation is a function of x: Draw a vertical line anywhere and cross the graph at most once, then it is a function. y= f(x) f(x) x x

Piecewise Function Graph -3

Domain/Range from Graph Look at graph to determine domain (inputs) and range (outputs). f(x) Domain: 2 Range: -2 2 4 x -2

12. Average Rate of Change Average Rate of Change (A.R.O.C.): change in the function values over the change in the input values. For a function, y = f(x), between x = a and x = b, the A.R.O.C is:

Example 1 Find the average rate of change between the indicated points. A.R.O.C. = y (-16, 6) 9 3 x -16 -12 -8 -4 4 8 12 16 (-4, -6) -9

Example 2 Find the average rate of change between the indicated points. A.R.O.C. = y (-16, 6) (12, 9) 9 3 x -16 -12 -8 -4 4 8 12 16 -9

Example 3 Find the average rate of change of the function between the values of the variable. A.R.O.C.

Example 4 Find the average rate of change of the function between the values of the variable. A.R.O.C.

Example 5 A man is running around a track 200 m in circumference. With the use of a stopwatch, his time is recorded at the end of each lap, seen in the table below. What was the man’s speed between 66s and 209s? Round answer to the nearest hundredth. Time (s) Distance (m) 32 200 66 400 104 600 153 800 209 1000 270 1200 341 1400 419 1600 Calculate the man’s speed for each lap. Please round answer to the nearest hundredth. Lap 1: Lap 2:

13. Combining Functions Take simple functions and combine for more complicated ones Arithmetic - add, subtract, multiply, divide Composition – evaluate one function inside another

Arithmetic Combinations Given the functions: Domain: Domain: Domain: Domain:

Composition

Example 1 Find the composition function: Domain:

Example 2 Find the composition function: Domain:

Example 3

Example 4 Use the graphs to evaluate: 2 -3 -1 4 -2

Application An airplane is flying 300 mi/hr at an altitude of 2 miles. At t = 0, the plane passes directly over a radar station. Express s as a function of d. d 2 s Express d as a function of t. Express s as a function of t.

14. Transformations Vertical Shifts f(x) + c: Add to the function => shift up f(x) - c: Subtract from the function => shift down

Horizontal Shifts f(x+c): Add to the variable => shift back(left) f(x – c): Subtract from the variable => shift forward(right)

Example: Shifts Graph the function:

y-axis reflection f(-x): Negate the variable => reflect in the y-axis

x-axis reflection -f(x): Negate the function => reflect in the x-axis

Example: Shift and reflection Graph the function:

Vertical Stretch & Shrink cf(x): multiply the function by c > 1=> vertical stretch (8,2) (1,1) cf(x): multiply the function by 0 < c < 1 => vertical shrink (2,4)

Even & Odd Functions Even function(y-axis symmetry): if f(-x) = f(x) Odd function(origin symmetry): if f(-x) = -f(x) Determine whether the following are even, odd, or neither.

15. Inverse Functions One-to-one functions: a function is one-to-one if every input is associated with one output and each output is associated with only one input. Horizontal Line Test – a function is one-to-one if and only if no horizontal line intersects the graph more than once.

Inverses Every one-to-one function, f(x), has an associated Function called an inverse function, f -1(x). The inverse function reverses what the function does. Its input is another function’s output. Its output is another function’s input. 3 .77 5 A B 4 -2 4

Example 1

Finding Inverses Graphically Inverses swap x and y coordinates. 2 -3 -1 4 -2

Finding Inverses Algebraically Three step process: Write the equation y = f (x). Solve the equation for x in terms of y. Swap the x and y variables. The resulting equation is y = f -1(x).

Example 2 Find the inverse function for:

Example 3 Find the inverse function for:

Inverse Property