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# What is a Function?

A function is **a special relationship** where each input has **a single output**. It is often written as \(f(x)\) where $x$ is the input value. Example: \(f(x) = \frac{x}{2}\) ("f of x equals x divided by 2"). It is a **function** because each input \(x\) has a single output.

# Vertical Line test

A test that can be used to determine** if a relation is a function**. A relation is a function if **there are no vertical lines** that intersect the **graph** at **more than one point. **Note that **all functions are relationships**, but **not all relationships are functions.**

# Domain and Range

If you are comfortable with domain and Range already in forms like Domain **X ∈ [1,5] or 1 ≤ X ≤ 5**, Range **Y ∈ ]2,10[ or 2 < X < 5** Then you can move on.

If you have any issues this might help:

\iframe{https://www.youtube.com/embed/96uHMcHWD2E}

\iframe{https://www.youtube.com/embed/-DTMakGDZAw}

## Composite Functions

The composition of two functions \(f(x)\) and \(g(x)\) such that function **g(x) is applied first** and function $f\left(x\right)$ is applied second is the function $f\circ g\left(x\right)$. It maps x to $f\left(g\right(x\left)\right)$ and we write:

$(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$

**Be careful** to read composite functions in the correct order,** read from right to left**. The function **closest to the x** is the** first function** to be executed, the result of which is given to the next.

## The domain of composite functions

The domain of the function **f∘g** is the set of all possible x values of g such that the respective g(x) belongs to the domain of f.

## Inverse functions

Now let us consider the function $f\left(x\right)=2x$.

${f}^{-1}\left(x\right)$ undoes the action of f(x) and therefore ${f}^{-1}\left(x\right)=x/2.$

It is enormously important that you do not think of it as "one over f". Note that $\frac{1}{f\left(x\right)}$ is written as $\left[f\right(x{)}^{-1}]$

On a graph, $f\left(x\right)$ and ${f}^{-1}$ are symmetric in the line y = x, meaning that if you reflect $f\left(x\right)$ in y=x then you obtain ${f}^{-1}$, and vice versa.

### Finding the inverse of functions ( ${f}^{-1}\left(x\right)$)

- Write $y=f\left(x\right)$.
- Then replace all the x's with y's, and the y's by x's.
- Solve until you obtain an expression $y=g\left(x\right)$, where $g\left(x\right)$ includes all components except y.

## The Reciprocal function

**except x≠0**) to $\frac{1}{x}$ and thus,

**the reciprocal function**and any multiple

**is self-inverse**(the inverse is the same as the original).

# Transformations of Functions and Graphs

There are 3 types of transformations:

## Translations

Variable | function | result | notes |

$h$ | $y=f(x-h)$ | Translating the graph towards the right/left | Right when $h$ is positive |

$v$ | $y=f\left(x\right)+v$ | Translating the graph upwards/downwards | Upwards when $v$ is positive |

\graph{{x^2+1},{(x-1)^2}}

## Scalings

Variable | function | result | notes |

$s$ | $y=s\left(f\right(x\left)\right)$ | Stretching/compressing the graph along the y-axis | Stretching when s > 1 |

$c$ | $y=f\left(cx\right)$ | Compressing/stretching the graph along the x-axis | Compressing when c > 1 |

\graph{{1.5x^2},{0.2x^2}}

## Reflections

function | result |

$y=-f\left(x\right)$ | Reflection across y-axis |

$y=f(-x)$ | Reflection across x-axis |

\graph{{y=x}}

## Review

Type of Transformation | Equation |

Vertical Translation | $y=f\left(x\right)+v$ |

Horizontal Translation | $y=f(x-h)$ |

Vertical Stretch | $y=sf\left(x\right)$ |

Horizontal Stretch | $y=f\left(cx\right)$ |

Vertical Reflection | $y=-f\left(x\right)$ |

Horizontal Reflection | $y=f(-x)$ |

It is also worth noting that more than one transformation can occur. For example, the function $y=2{x}^{2}+3$ can be viewed as the basic quadratic function $y={x}^{2}$ **stretched vertically by a factor of 2** and **shifted vertically by a factor of 3**.

\graph{{4x^2+2}}

# Editors

- CD_FER - 805 words.
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