 # 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.

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