We intuitively say that a variable is a **function** of a second variable when its value depends on the value of the second variable and the value of the first variable can be *uniquely* calculated by some **rule** when the value of the second variable is assumed. The first variable is called the** dependent variable** and the second the **independent variable**.

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For example, the temperature at which pure water boils is a function of the altitude above sea level. The area of a circle $A$ is a function of its radius $r$, because if the radius of a circle $r$ is given, we can calculate its area $A=\pi r^{2}$. Thus the area is a function of the radius. Conversely, if the area of a circle is given, we can calculate its radius through $r=\sqrt{A/\pi}$. Thus, the radius is a function of the area. Sometimes, like in this example, it is a matter of choice which variable is called the independent variable and which one the dependent variable.

In a triangle, suppose the lengths of two sides, $a$ and $b$, are given. If we choose a value between $0$ and $180^{\circ}$ for the angle $\gamma$ between these two sides, then the length of the third side $c$ is determined. Thus if $a$ and $b$ are given, we can say $c$ is a function of $\gamma$. From geometry, we know that this function is represented by the formula

\[c=\sqrt{a^{2}+b^{2}-2ab\cos\gamma}.\]

The federal tax rate for a single person is a function of his or her taxable income. For example, if his or her taxable income is any number between $\$38,701$ and $\$82,500$, the tax rate is 22% and if the taxable income is between $\$9,526$ and $\$38,700$, the tax rate falls into 12%.

- A variable can be a function of more than one other variable. For example, the volume of a circular cylinder $V$ is a function of the radius of its base $r$ and the height of the cylinder $h$. We need to know both $r$ and $h$ to be able to calculate $V$ through $V=\pi r^{2}h$. We will study multivariable functions only in the second part of the course.

The formula

\[

y=\sqrt{x},

\]
defines $y$ as a function of $x$ for all $x\geq0$ (we restrict the values of $x$ to nonnegative numbers because we cannot take the root of negative numbers).

If $x$ and $y$ are two variables connected by the relation

\[

y=x^{2},

\]
then $y$ is a function of $x$ because if the value of $x$ is given, the value of $y$ can be calculated uniquely. Conversely if the value of $y>0$ is given (for example $y=4$), this equation defines two corresponding values of $x$ (for example, $x=+2$ or $x=-2$ corresponding to $y=4$). Because not a unique value of $x$ corresponds to a given value of $y$ (except when $y=0$), $x$ is not a function of $y$.

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### Notation

In mathematics, we often wish to refer to a generic function without specifying any particular formula, table, or graph. To denote that $y$ is a function of $x$, we write

\[

y=f(x),

\]
which is read as “$y$ is equal to $f$ of $x$.” In this notation,$f$ represents the function, that is, the “rule” or “procedure” (which usually but not always involves some formula) associating the values of $x$ to the values of $y$. Instead of $f(x)$, we may use other notations such as $g(x),\phi(x),F(x),f'(x),s(x)$, etc. If more than a function occur in a problem, one may be expressed as $f(x)$, another as $F(x)$, another as $g(x)$, and so on. It is also convenient in practice to represent different functions by the symbols $f_{1}(x),f_{2}(x),f_{3}(x)$, etc. However, during any investigation, the same functional symbol always indicates the same law of dependence of the dependent variable upon the independent variable.

### Visualizations

A function can be thought of as a machine or a computer program that assigns one output to every allowable input.

(a) | (b) |

**Figure 1:** A function can be thought of as a (a) machine or (b) computer program that for each allowable input gives one output.

Another way to picture a function is by an arrow diagram. For each element $x$ in $A$, the value of $f(x)$ in $B$ is to be found at the head of the corresponding arrow. As we can see in Figure 2(a), $f$ associates to each element in $A$, one and only one element in $B$. Thus, $f$ is a function, although two elements in $A$ are associated with one element in $B$. But in Figure 2(b), $g$ associates two elements to an element in $A$. Therefore, $g$ is not a function.

(a) | (b) |

**Figure 2:** $f$ is a function, but $g$ is not because $g$ assigns two elements in its codomain to $\star $ in its domain.

### Definitions

**Definition:** A function $f$ from a set $A$, to a set $B$, is a rule that assigns, to each element $x$ in $A$, **one**

**and only one **element $y$ in $B$. We then write $y=f(x)$.

Sets $A$ and $B$ are called **domain** and co-domain of $f$, respectively. To mention that $f$ is a function with domain $A$ and co-domain $B$, we write $f:A\rightarrow B$.

- If $y=f(x)$, we also call the independent variable $x$, the
**argument**of the function, and the element $y$**the value of $\boldsymbol{f}$ at $\boldsymbol{x}$**or the**image of $\boldsymbol{x}$ under $\boldsymbol{f}$.** - The definition of a function $f:A\to B$ does not restrict the nature of the elements of $A$ and $B$, but in elementary calculus we assume that they are numbers; that is, $A$ and $B$ are subsets of real numbers $\mathbb{R}$, unless otherwise stated.

### Formula of the Function

If $y=f(x)$, the particular value of the function when $x$ has a definite value $a$ is then expressed as $f(a)$. For example, if

\[

f(x)=4x^{2}-5x+1,

\]
then

\[

f(-1)=4(-1)^{2}-5(-1)+1=10,

\]
and

\[

f(0)=4(0)^{2}-5(0)+1=1.

\]
Also, because it does not matter which letter we use for the independent variable, we have

\[

f(t)=4t^{2}-5t+1,

\]
\[

f(u)=4u^{2}-5u+1,

\]
\[

f(b+1)=4(b+1)^{2}-5(b+1)+1=4b^{2}+3b.

\]

Strictly speaking $f(x)$ is the value of $f$ at $x$, but we often talk about “the function $f(x)$” or “the function $y=f(x)$.” For example, consider a function $f:\mathbb{R\to\mathbb{R}}$ that takes a number $x$ and gives its square $x^{2}$. In this case, we can simply say:

1.“the function $f(x)=x^{2}$;”

2.“the function $y=x^{2}$” (if we denote the dependent variable by $y$);

3. simply “the function $x^{2}$.”

We can also connect the input and output values by a special arrow, namely “the function $x\mapsto x^{2}$.”

## Implicit and explicit functions

If an equation between several variables is solved for any one, the latter is said to be an explicit function of the others, the manner of its dependence being exhibited by the solution of the equation. Otherwise, it is said to be an implicit function. Thus, in $x^{2}+y^{2}=4,$ $y$ is an implicit function of $x$; while, in $y=\sqrt{4-x^{2}},$ $y$ is an explicit function of $x$. The difference is one of form only. The notation $y=f(x)$ is used to denote that $y$ is an explicit function of $x$, and the notation $f(x,y)=0$ to denote that $x$ and $y$ are implicit functions of each other.