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Operations with Functions

Composition of functions

Let $f$ and $g$ be two functions in $x$, where $x$ ∈ ℝ, then:

1. $(f±g)(x) = f(x) ± g(x)$

2. $a(f(x)) = af(x)$, $a$ ∈ ℝ

3. $(fg)(x) = f(x)g(x)$

4. $(f/g)(x) = {f(x)}/{g(x)}$, $g(x)$ ≠ 0

The composition of two functions, g and h, such that g is applied first and h second, is given by:

(g o h)(x) = g(h(x))

The domain of the composite function (g o h) is the set of all x in the domain of h such that h(x) is in the domain of g.

Example

e.g. Let f(x) = √x, and g(x) = $x^2$.

(f o g)(x) = f(g(x)) = $√{x^2} = x$

(g o f)(x) = g(f(x)) = $(√x)^2 = x$

e.g. Let $f(x) = √{x+4}$, and g(x) = $x^2$.

(f o g)(x) = f(g(x)) = $√{(x^2+4)} ≠ x + 4$

(g o f)(x) = g(f(x)) = $(√{x+4})^2 = x + 4$

From these examples, we can see that (f o g)(x) is not always equal to (g o f)(x)!

Decomposing Composite Functions

The function that is applied first is the 'inside' function, and that applied second the 'outside' function. e.g. in $f(x) = (x+3)^2 = (g(h(x))$, the inside function is $h(x) = x+3$, and the outside function is $g(x) = x^2$.

Domains of Composite Functions

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Composite functions have restricted interdependent domains

If x is in the domain of the composite function (g o h), then x must be in the domain of h, and h(x) must be in the domain of g.

The inverse of a function $f(x)$ is written as $f^{-1}$. The effect of the inverse of a function is to reverse the action of the function: if x undergoes a transformation to y through function $f(x)$, then applying $f^{-1}(x)$ to y will return the original value of x.

For example, if $f(x) = 2x + 4$, then $f^{-1}(x) = {x - 4}/2$.

Not all functions have an inverse. Be careful, $f^{-1}(x)$ is not the same as $[f(x)]^{-1}$. $f^{-1}(x)$ is a reflection across the line $y = x$, and $[f(x)]^{-1}$ is a reflection across the x-axis.

Functions that do not have an inverse can be tested by the horizontal line test.

If any horizontal line crosses the graph of a function more than once, then the function has no inverse. This is because the inverse is not a function, but a relation (2 values of y for one value of x.

An example of a function that has no inverse is $f(x) = x^2$, while $f(x) = x^3$ does have an inverse.

Inverse function
An inverse function is a reflection across the line y = x

Finding inverses algebraically

If $f(x) = 4x - 6$, the inverse can be found by setting y as $f(x)$, then exchanging the places of x and y:

$y = 4x - 6$, so $x = 4y - 6$ for the inverse.

Rearranging to return y to the left:

$y = {x + 6}/4$

This can be written as the inverse: $f^{-1}(x) = {x + 6}/4$

To check that a function is the inverse of another function, combine the functions:

$$(f o f^{-1})(x) = x$$

Example above: $(f o f^{-1})(x) = 4({x + 6}/4) - 6 = x$

A self-inverse function is a function whose inverse is the same. e.g. $f(x) = a/x$ and $f(x) = a - x$, where a ∈ ℝ.

Identity Function

$$(f o f^{-1}) = I$$

where $I(x) = x$. The identity function leaves $x$ unchanged.

Content © Andrew Bone. All rights reserved. Created : September 26, 2014 Last updated :July 20, 2015

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