Chapter 4: Functions

Chapter 4: Functions

1. Reflecting Graphs; Symmetry

2. Operations of Functions

   • Sum of f and g: (f+g)(x)=f(x)+g(x) Difference of f and g: (f-g)(x)=f(x)-g(x) Product of f and g: (f*g)(x) = f(x)*g(x) Quotient of f and g: (f/g)(x)= f(x)/g(x)
   • Composite of functions
     • denoted f o g is defined by 2 conditions
       • 1. (f o g )(x)=f(g(x)), which means f of g of x. 2. x is in the domain of g and g(x) is in the domain of f.

3. Properties of Functions

   • A function assigns every element in the Domain set (x) to exactly one element in the Rang set (y)
     • You can use the vertical line test to see if it is a function - If there are two intersections on a single vertical line by the graph, then it is not a function.
       • The Zeroes of y= f(x) are the x-intercepts of the graph

4. Reflecting Graphs; Symmetry

   • An Axis of Symmetry of a graph is a line that acts as a perpendicular bisector of the graph
   • 1. Reflection in the x-axis: The graph of y=-f(x) is obtained by reflecting the graph of y=f(x) in the x-axis. 2. Reflection when the graph of y=f(x) becomes y=|f(x)|, it means that the parts that dip below the x-axis in the previous graph, bends upward so that none crosses into the negative y-axis. 3. Reflection in the y-axis: the graph of y=f(-x) is obtained by reflecting the graph of y=f(x) in the y-axis. 4. Reflection on the line y=x --> you interchange the x and y coordinates

5. Periodic Functions; Stretching and Translating Garphs

   • The function f is periodic if there is a positive number p, called a period of f, such that f(x+p)=f(x) for all x in domain of f. The smaller period of the function is called the fundamental period
   • The Amplitude of a periodic function is given by: A= (M-m)/2 (for a fundamental period with highest point M and lowest point m)
   • Vertical Shrink: when y= f(x) becomes y= (1/2)f(x). Vertical Stretch: when y=f(x) becomes y=2f(x)
     • Horizontal Shrink: when y=f(x) becomes y=f(2x) Horizontal Stretch: when y= f(x) becomes y= f(1/2 x)
   • Translating Graphs
     • The graph of y-k=f(x-h) is obtained by translating the graph of y=f(x) horizontally by h units and vertically by k units.

6. Inverse Functions

   • in order to find the inverse function of y=f(x), swap the y and x coordinates and solve for y. An inverse is always a 1-to-1 function-- each y value corresponds to only 1 x value. (Thus, we must use the Horizontal Line test in order to determine that).

7. Forming Functions form Verbal descriptions

   • Minimum and Maximum values: in many situations, a problem is modeled by a function of more than one variable. By expressing one variable in terms of another, a function of 2 variables can be transformed into a function of one variable.

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