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Functions is an important branch of math, which connects the variable x with the variable y. Functions are generally represented as y = f(x) and it states the dependence of y on x, or we say that y is a function of x. Functions formulas define the mathematical rules to connect one set of elements to another set of elements. Numerous operations of functions can be easily performed with the help of these function formulas.
What is the list of Function Formulas?
The list of function formulas can be broadly classified as formulas for performing the numerous arithmetic operations across functions, and for performing the combined operations involving two or more functions.
Formula 1 The arithmetic operation of addition, subtraction, multiplication, and division across functions can be conveniently performed similarly to the normal arithmetic operations across numbers. Here f(x) and g(x) and the two functions across which the following arithmetic operations are performed.
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(αf)(x) = αf(x)
(fg)(x) = f(x).g(x)
(f/g)(x) = f(x)/g(x)
Formula 2 Composite function properties includes two or more functions within a formula. The function f(x), g(x), h(x) are combined to form composite functions gof(x), hog(x), ho(gof)(x).
gof(x) = g(f(x))
ho(gof)(x) = (hog)of(x)
(gof)-1(x) = f-1og-1(x)
Solved Examples on Function Formulas
Example 1: Find gof(x), given that f(x) = 2x + 1 and g(x) = x - 1.
Given f(x) = 2x + 1, and g(x) = x - 1
Applying the function formulas for gof(x) we have.
gof(x) = g(f(x)) = g(2x + 1) = (2x + 1) - 1 = 2x + 1 - 1 = 2x
Answer: gof(x) = 2x
Example 2: Find the inverse of the function f(x) = 3x + 5, through the application of function formulas.
Given f(x) = 3x + 5
f(x) - 5 = 3x
3x = f(x) - 5
x = (f(x) - 5)/3
f-1(x) = (x - 5)/3
Answer: f-1(x) = (x - 5)/3