Chapter 2 Functions

2.1 What is a Function? The idea of a function is one of the most basic and important ideas in mathematics. Nearly all physical phenomenon has one quantity dependant upon another.

Functions vs. Relations A relation is a correspondence between the domain and range such that each member of the domain corresponds to at least one member of the range. A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

Function Notation Functions used in math are given by equations. Consider: To get ordered pairs, we “input” values for x and values for y are “output” The “input” and “output” are grouped as an ordered pair.

Function Notation A more concise notation is used. We name the function, typically “f”. We name the inputs “x” and the outputs “y”. f(x) is read “f of x” or “f evaluated at x” or “ f at x” We replace y with f(x):

Using Function Notation “Evaluate when x = 3” is now written as f(3). “Evaluate when x = -3” is now written as f(-3)

Piece-wise defined functions Your x value will determine which function to use. Always evaluate the endpoints in each of the functions. Check the inequality symbol to determine if the endpoint is included or not.

Finding Domain Algebraically The domain consists of values of x for which the function is defined. To find the domain, we find values of x for which the function is not defined. Two questions: 1. Are there any variables in the denominator? 2. Are there any variables in the radicand?

Difference Quotient The difference quotient is very important in the study of Calculus. Remember that EVERYTHING in parentheses must be inserted wherever there is an x in the function.

2.2 Graphs of Functions Remember when graphing functions, we can always plot points. Input values are the x coordinates, and the output values are the y coordinates. If unsure about the shape of the graph, plot at least 5 points, using both positive and negative x values, and zero. Don’t forget you can always use x and y intercepts.

The Vertical Line Test To determine if a graph represents a function, we use the Vertical Line Test. If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function.

2.3 Applied Functions: Variation Direct Variation: Read “y varies directly as x” or ‘y is directly proportional to x”. k is the constant of variation. Inverse Variation: Read “y varies inversely as x” or “y is indirectly proportional to x”. k is the constant of variation.

Combined Variation “y varies directly as the nth power of x.” “y varies inversely as the nth power of x.” “y varies jointly as x and z”

2.4 Average Rate of Change Definition: The average rate of change of the function between and is average rate of change =

y x

Increasing and Decreasing Functions On a given interval, if the graph of a function rises from left to right, it is said to be increasing. If the graph drops from left to right, it is said to be decreasing. If the graph stays the same from left to right, it is said to be constant.

Increasing function A function f is said to be increasing on an open interval I, if for all a and b in I, a < b implies

Decreasing Function A function f is said to be decreasing on an open interval I, if for all a and b in I, a < b implies

Constant Function A function f is said to be constant on an open interval I, if for all a and b in I,

2.5 Transformations of Functions VERTICAL SHIFTING Shift up c units. c Add c to the y coordinate.

Subtract c from the y coordinate. VERTICAL SHIFTING Shift down c units. Subtract c from the y coordinate. c

Add c to the x coordinate. HORIZONTAL SHIFTING Shift right c units. c Add c to the x coordinate.

Subtract c from the x coordinate. HORIZONTAL SHIFTING Shift left c units. c Subtract c from the x coordinate.

REFLECTING GRAPHS Reflection in the x axis. Change the sign of the x coordinate.

REFLECTING GRAPHS Reflection in the y axis. Change the sign of the y coordinate.

VERTICAL STRETCHING AND SHRINKING OF GRAPHS Stretch the graph vertically by a factor of a. The graph will be taller than the original. Multiply the y coordinate by a. Shrink the graph vertically by a factor of a. The graph will be shorter than the original. Multiply the y coordinate by a.

HORIZONTAL STRETCHING AND SHRINKING OF GRAPHS Shrink the graph horizontally by a factor of 1/a. The graph will be narrower than the original graph. Divide the x coordinate by a. Stretch the graph horizontally by a factor of 1/a. The graph will be wider than the original graph. Divide the x coordinate by a.

Applying Transformations Use order of operations. Divide x-coordinate by c. Add d to x coordinate if “-d”. Subtract if “+d”. Multiply y-coordinate by a. Add b to y-coordinate if “+b”. Subtract if “-b”.

Even and Odd Functions If a function is symmetric with respect to the y – axis, the function is said to be an even function. Replace x with –x. If the original results, then the function is even. If a function is symmetric with respect to the origin, the function is said to be an odd function. Replace x with –x. If the opposite of the original function results, then the function is odd. Can a function have symmetry with respect to the x – axis?

2.6 Extreme Values of Functions A second degree function, or a quadratic function, has a graph called a parabola. The important features of a parabola are its vertex, x and y intercepts, and its line of symmetry.

Standard Form of a Quadratic Function A quadratic function can be expressed in the standard form by completing the square. Vertex (h,k). Opens up if a>0 and opens down if a<0. Maximum value at vertex if a<0, and the max value is Minimum value at vertex if a>0, and the min value is

Complete the Square to Graph Move constant to other side. Divide 10 in half, adding its square to each side. Factor. Bring constant back to the original side.

Identifying Important Aspects from this Form of the Equation The vertex is (-5, -2), the line of symmetry is x=-5 and the graph opens up. The y intercept is (0, 23). To find the x intercepts, solve

Maximum or Minimum Value of a Quadratic Function The maximum or minimum value of a quadratic function occurs at If a>0, then the minimum value is If a<0, then the maximum value is

Local Extreme Values Suppose that f is a function for which f(c) exists for some c in the domain of f. Then: f(c) is a relative maximum if there exists an open interval I containing c such that f(c)>f(x) for all x in I where x is not equal to c. f(c) is a relative minimum if there exists an open interval I containing c such that f(c)<f(x) for all x in I where x is not equal to c.

Steps for Finding Extreme Values on Calculator Press CALC (2nd then TRACE). Arrow down to highlight 3:minimum. Press enter. Move cursor to left of minimum and press enter to set the left bound. Move cursor to the right of minimum, press enter to set right bound. Move cursor anywhere in between for a guess. Hit enter. The calculator then shows the minimum.

Now find max. Press CALC. Arrow down to highlight 4:maximum. Press enter. Move cursor to left of max, press enter. Move cursor right of max, press enter. Move cursor anywhere in between for a guess. The calculator then shows the maximum.

Steps for Finding Extreme Values on Calculator Press CALC (2nd then TRACE). Arrow down to highlight 3:minimum. Press enter. Move cursor to left of minimum and press enter to set the left bound. Move cursor to the right of minimum, press enter to set right bound. Move cursor anywhere in between for a guess. Hit enter.

Now find max. Press CALC. Arrow down to highlight 4:maximum. Press enter. Move cursor to left of max, press enter. Move cursor right of max, press enter. Move cursor anywhere in between for a guess.

2.7 Modeling with Functions Five Steps for Problem Solving Read the problem. Identify what you are looking for. Define variables. Set up the equation. Solve the equation. Check. Is your answer reasonable? Write the answer using a complete sentence.

2.8 Combining Functions Blue box on Page 218. All operations work as they do with numbers. When considering the domain of a quotient of two function, be sure to remove any domain values that would yield a zero in the denominator.

Composition of Functions Given two functions f and g, the composite function (also called the composition of f and g) is defined by Example: Given and find … a. and their domains. The domain for each is all real numbers.

Decomposing a Function as a Composition An important skill needed in calculus is to be able to recognize how a function can be expressed as a composition of two functions. The easiest way to do this is to determine and “outside” function and an “inside” function.

2.9 One-to-One Functions and Their Inverses Definition: A function within domain A is called a one-to-one function if no two elements of A have the same image, that is, whenever . OR If then .

Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once. If it is possible for a horizontal line to intersect the graph of a function more than once, then the function is not one-to-one and its inverse is not a function.

Inverse Function Definition: Let f be a one-to-one function with domain A and range B. Then its inverse function has domain B and range A and is defined by for any y in B.

Inverse Functions and Composition If a function is one-to-one, then is the unique function such that each of the following holds: for each x in the domain of f, and for each x in the domain of .

Steps for Finding an Inverse Replace f(x) with y. Swap x and y. Solve for y. Replace y with .

Graphs of Functions and their Inverses The graph of is obtained by reflecting the graph of f in the line y=x.