 # 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 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. 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.

## Site Index

### Latest Item on Science Library:

The most recent article is:

Trigonometry

View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

### Physics

Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on ScienceLibrary.info. ### Great Scientists

#### Gro Harlem Brundtland

b. 1939

Gro Harlem Brundtland, born 1939, is one of the most important figureheads and leaders of the environmental movement. In 1987, she chaired the famous Brundtland Commission for the UNO, which established Sustainable Development as a central goal of development and environmental protection.  ### Quote of the day... When I said I wanted to be a comedian, they laughed. Well, they're not laughing now... 