# operations with functions

We define the **addition, subtraction, multiplication and division of functions** as:

· (f ± g)(x) = f(x) ± g(x)

· (f · g)(x) = f(x) · g(x)

· (f/g) (x) = f(x)/g(x) (if g(x)≠0)

Example: if f(x) = x^{2} -2 and g(x) = 3x + 2, then:

· (f + g)(x) = f(x) + g(x) = x^{2} + 3x

· (f - g)(x) = f(x) - g(x) = x^{2} – 3x - 4

· (f · g)(x) = f(x) · g(x) = 3x^{3} + 2x^{2} – 6x - 4

· (f/g) (x) = f(x)/g(x) = (x^{2} -2)/(3x + 2), if x ≠ -2/3

**Function composition**is the application of one function to the results of another. It is represented by g

_{°}f, and we say “f composed with g”:

g_{°}f(x) = g(f(x)) (if f(x)Є Dom g)

^{2}, then:

g_{°}f(x) = g(f(x)) = g(x + 1) = (x + 1)^{2} = x^{2} + 2x +1

f_{°}g(x) = f(g(x)) = f(x^{2}) = x^{2} + 1

g_{°}f ≠ f_{°}g

**inverse function**of f is a function that undoes another function, that is, it is a function f

^{-1}such that f

_{°}f

^{-1}(x) = f

^{-1}

_{°}f (x) = i(x) = x

^{2}, then f

^{-1}(x) = √x, because

f_{°}f^{-1}(x) = f(√x) = (√x)^{2}= x

f^{-1} _{°}f (x) = f^{-1}(x^{2}) = √x^{2} = x

^{-1}(x) = 1/x, because

f_{˚} f^{-1}(x) = f(1/x) = 1/(1/x)= x

f^{-1} _{˚}f (x) = f^{-1}(1/x) = 1/(1/x)= x

Example 3: find the inverse function of f(x) = √(2x)

x = √(2y) → x^{2} = 2y → y = f^{-1}(x) = x^{2}/2

NOTE: inverse functions are symmetric and their axis of symmetry is the line: y = x

**Exercise:**

Solutions:

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