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**>> 1.8. Combinations of Functions **

1.7. Transformations of Functions

There are several useful ways of transforming functions, including * reflections*,

*, and*

**translations***. First we consider reflections. For a given function f(x), consider the function g(x) = -f(x). This is simply the negative of f(x). In other words, for every image y of x under f, -y is the corresponding image under g = -f. As a result, the graph of g is the reflection of f about the x-axis. What about h(x) = f(-x)? If y is the image of x under f, then it is easy to see that y is the corresponding image of -x under h. The graph of h is the reflection of f about the y-axis.*

**scale transformations**Some functions remain unchanged when reflected about the y-axis. In other words, these functions are symmetric about the y-axis. Such functions are called * even functions*. Mathematically speaking, a function f(x) is even if f(-x) = f(x) for all x in the domain of f. The reason for this name is that a polynomial f(x) is even if and only if all its nonzero coefficients correspond to even powers of x, i.e. if and only if f(x) has the form f(x) = a

_{2n}x

^{2n}+ a

_{2n-2}x

^{2n-2}+ ... + a

_{4}x

^{4}+ a

_{2}x

^{2}+ a

_{0}for some nonnegative integer n. The absolute value function f(x) = |x| is another example of an even function.

An equally important class of functions are * odd functions*. These are functions f(x) satisfying f(-x) = -f(x) for all x in the domain of f. Odd functions are symmetric about the origin, i.e. if the point (x,y) lies on the graph of f, then the point (-x, -y) also lies on the graph of f. The reason for this name is that a polynomial f(x) is odd if and only if all its nonzero coefficients correspond to odd powers of x, i.e. if and only if f(x) has the form f(x) = a

_{2n+1}x

^{2n+1}+ a

_{2n-1}x

^{2n-1}+ ... + a

_{5}x

^{5}+ a

_{3}x

^{3}+ a

_{1}x for some nonnegative integer n.

It turns out that every function defined over all real numbers has a unique decomposition into even and odd parts. The decomposition is as follows. Given a function f(x) defined over the reals, let f_{e}(x) = [f(x) + f(-x)] / 2 and
f_{o}(x) = [f(x) - f(-x)] / 2. Then it is completely straightforward to verify that f_{e} is even, f_{o} is odd, and that f = f_{e} + f_{o}. We leave this as an exercise.

Given a function f(x) and a real constant a, consider the function g(x) = f(x - a). If y = f(x) is the image of x under f, then y = g(x + a) is the image of x + a under g. In other words, g acts the same way on every input x+a as f acts on x. As a result, the graph of g looks the same as that of f, but shifted by a distance a along the x-axis.

Given a function f(x) and a real constant b, consider the function g(x) = f(x) + b. If y = f(x) is the image of x under f, then y + b = g(x) is the image of x under g. The graph of g looks the same as that of f, but shifted by a distance b along the y-axis.

Given a function f(x) and a positive real constant c, consider the function g(x) = f(x/c). If y = f(x) is the image of x under f, then y = g(cx) is the image of cx under g. In other words, g acts the same way on every input cx as f acts on x. As a result, the graph of g looks the same as that of f, but stretched out by a factor of c along the x-axis. (If c is less than 1, then the graph of g is compressed along the x-axis.)

Given a function f(x) and a positive real constant d, consider the function g(x) = d f(x). If y = f(x) is the image of x under f, then dy = g(x) is the image of x under g. The graph of g looks the same as that of f, but stretched out by a factor of d along the y-axis. (If d is less than 1, then the graph of g is compressed along the y-axis.)