# Functions as Transformations

You have learned that a function is a relation which has only one output for each input. This means that a function can be thought of as a machine or rule that transforms any element from its domain into a single element of its range. One way to get a function is by evaluating an algebraic expression containing one variable, such as \$x\$. You can also build functions by composing other functions. You will learn about these methods in this lesson.

## Functions defined by a table, formula, or graph

A function can be thought of as a machine or rule that transforms an input to an output. If \$f\$ is a function, we write \$f(x)\$ (read as “\$f\$ of \$x\$”) for the output value that \$f\$ produces when given the input \$x\$.

The function \$f\$ is given in the table to the right, and graphed on the grid to the left.

Notice that for this function \$f(1)=-3\$. That is, \$f\$ transforms \$1\$ into \$-3\$. What is \$f(3)\$? (This is read as “\$f\$ of 3.”)

When \$x\$ is not in the domain of \$f\$ (that is, there is no \$f(x)\$-value for the input \$x\$), we say that \$f(x)\$ “is not defined.” For example, \$f(0)\$ is not defined. What is another value of \$x\$ where \$f(x)\$ is not defined?

\$x\$\$f(x)\$

A function can also be specified by a formula. For example, we might define a function \$g\$ by saying that \$g(x)=x^2-2\$. This tells you that \$g\$ transforms each number \$x\$ into the number \$x^2-2\$. The function \$g\$ is graphed on the grid to the left.

If \$g(x)=x^2-2\$, then \$g(1)=1^2-2=1-2=-1\$. What is \$g(2)\$?

What is \$g(-1)\$?

Another way to specify a function is by giving its graph. For example, the function \$h\$ is graphed on the grid to the left. If you drag the vertical bar on the grid to \$x=1\$, you can see that \$h(1)=-4\$.

 Drag the vertical bar to \$x=-1\$. What is \$h(-1)\$?
 What is \$h(2)\$?

## Composing functions

You can build a new function out of two old functions by first doing one of them to the input, and then doing the other one to the result. If \$f\$ and \$g\$ are two functions, their composition is the function which transforms \$x\$ into \$f(g(x))\$ (“\$f\$ of \$g\$ of \$x\$”). That is, we first transform \$x\$ using the function \$g\$, to get \$g(x)\$, and then transform \$g(x)\$ using the function \$f\$, to get \$f(g(x))\$. We also write this combined function as \$f ∘ g\$ (read as “\$f\$ composed with \$g\$”).

In this question, we’ll keep using the functions \$f\$, \$g\$, and \$h\$ defined in the previous section. The function \$f\$ is given by the table to the right, the function \$g\$ is given by the formula \$g(x)=x^2-2\$, and the function \$h\$ is graphed to the left.

To compute \${f ∘ g}(1)\$, start by writing it as \$f(g(1))\$. This can be computed from the inside out: first, notice that \$g(1)=1^2-2=-1\$. So \$f(g(1))=f(-1)\$. Now, you can look at the table to see that \$f(-1)=1\$. So \${f ∘ g}(1)=f(g(1))=f(-1)=1\$.

\$x\$\$f(x)\$

Compute \${g ∘ f}(1)\$.

 Is \${g ∘ f}(1)\$ equal to \${f ∘ g}(1)\$?
 Can you compose functions in any order, and always get the same result? (That is, are \$g ∘ f\$ and \$f ∘ g\$ the same function?)
 The function \$h\$ is still graphed on the grid to the left. What is \${h ∘ f}(4)\$?

If you have two functions that are both defined by a formula, their composition can also be defined by a formula. For example, if the functions \$g\$ and \$k\$ are defined by the formulas \$g(x)=x^2-2\$ and \$k(x)=x+3\$, then \$\cl"red"{k ∘ g}\$ is defined by

\$\$ {k ∘ g}(x) = k(g(x)) = k(x^2-2) = x^2 - 2 + 3 = x^2 + 1 \$\$

What is a formula for \$g ∘ k\$?

Click to see the graphs of \$\cl"red"{k ∘ g}\$ and \$\cl"green"{g ∘ k}\$.

 Are the graphs of \$\cl"red"{k ∘ g}\$ and \$\cl"green"{g ∘ k}\$ the same or different?
 Is \$\cl"red"{k ∘ g}\$ the same function as \$\cl"green"{g ∘ k}\$?

If \$\cl"red"{f}\$ and \$\cl"blue"{g}\$ are functions defined by the formulas \$\cl"red"{f(x)=2x+1}\$ and \$\cl"blue"{g(x)=3x-4}\$, find a formula for \$f ∘ g\$.

Just as equations of the form \$y=mx+b\$ are called linear equations, functions defined by a formula of the form \$f(x)=mx+b\$ are called linear functions.

 Are \$\cl"red"{f}\$ and \$\cl"blue"{g}\$ both linear functions?
 Is \$f ∘ g\$ a linear function?
 If \$\cl"green"{h}\$ and \$\cl"purple"{k}\$ are functions defined by the formulas \$\cl"green"{h(x)=x+2}\$ and \$\cl"purple"{k(x)=4x-3}\$, use scratch paper to find a formula for \$h ∘ k\$.
 Notice that \$\cl"green"{h}\$ and \$\cl"purple"{k}\$ are both linear functions. Is \$h ∘ k\$ a linear function?

In fact:

If \$f\$ and \$g\$ are linear functions, their composition \$f ∘ g\$ is also a linear function.