Function Notation and Function Operations

The equation that defines a function f can be expressed in the form y=f(x). Thus, y and f(x) are two names for the same quantity – the element that is paired off with x. The notation f(x) is read as “f of x” and called the value of the function f at x. With this notation, you don’t have to worry about which function a given ordered pair (x, y) belongs to. If y=f(x), it belongs to the function f; if y=g(x), then it belongs to another function g. To find the value of the function, simply substitute the value in the parentheses in place of x.

Example:If f is defined by f(x) = x2-3, then f(-1) = (-1)2-3 = -2. This means that (-1,-2) is an element of this function.

Function OperationsIf f and g are two functions, then the sum f+g is defined by y=(f+g)(x)=f(x)+g(x). If f(x)=x2-1 and g(x)=3x+2, what is the value of (f+g)(x)? of (f+g)(-1)? If your answers are(f+g)(x) = f(x)+g(x) = (x2-1)+(3x+2) = x2+3x+1(f+g)(-1) = (-1)2+3(-1)+1=-1then you are correct. Can you give similar definitions for the difference f-g, the product fg, and the quotient f/g of the two functions f and g?

If f and g are two functions defined by y=f(x) and y=g(x) respectively, thena. their difference f-g is defined by y = f(x)-g(x);b. their product fg is defined by y = f(x)·g(x);c. their quotient is defined by , if g(x)≠0.

The notation g(f(x)) is read as “g of f of x”. In finding the value of the composition of g and f, the inner function f is applied first by taking a value x and evaluating f(x) to get an element z in Z. Then outer function g is applied next by computing y=g(z)=g(f(x)). The ordered pairs that belong to the composition are of the form (x, y).The composition of f and g, denoted by f o g, may be defined in a similar manner. Can you form this definition? You may rename the ordered pairs in f and g so that the starting variable is always x, and the ending variable is y.

For each of the given pairs of functions, find f+g, f-g, fg, and .1.2.3.

Exercises: