Chapter 1: Functions

A

is a correspondence that associates with each element*function f*of a set*a*__called the__one and only one element*domain*of a set*b*__called the__*range**.*

We write ** f**(

**Inputs**- are the elements in the*domain*.**Outputs**- are the elements in the*range*.A

**function**is often represented by an equation, a graph, or a table.A

**vertical line**cuts the graph of a function in at most one point

**Example :**

Two functions

*f*and*g*with the same domain may be combined**to yield their sum and difference**:

*f*(*x*) + *g*(*x*) and *f*(*x*) − *g*(*x*), also written as (*f* + *g*) (*x*) and (*f* − *g*) (*x*)

or their product and quotient: *f*(*x*)*g*(*x*) and *f*(*x*)/*g*(*x*), also written as (*fg*) (*x*) and (*f* /*g*) (*x*)

The quotient is defined for all

*x*in the shared domain except those values for which*g*(*x*), the denominator, equals zero.

**Example:**

The

(or*composition**composite*) of*f*with*g*, written as*f*(*g*(*x*)) and read as “*f*of*g*of*x*,” is**the function obtained by replacing**wherever it occurs in*x**f*(*x*) by*g*(*x*). We also write (*f*∘*g*) (*x*) for*f*(*g*(*x*)).The domain of (

*f*∘*g*) (*x*) is the set of all*x*in the domain of*g*for which*g*(*x*) is in the domain of*f*.

**Example:**

**SOLUTION**:

The *absolute-value***function** *f*(*x*) = |*x*| and the *greatest-integer***function** *g*(*x*) = [*x*] are sketched in the Figure below:

**Example 8**

A function *f* is defined on the interval [−2,2] and has the graph shown in Figure 1-4.

(a)Sketch the graph of *y* = |*f*(*x*)|.

(b)Sketch the graph of *y* = *f*(|*x*|).

(c)Sketch the graph of *y* = −*f*(*x*).

(d)Sketch the graph of *y* = *f*(−*x*).

SOLUTIONS:

A

is of the form*polynomial function*

where ** n** is a

If *an* ≠ 0, the degree of the polynomial is *n*.

A

*linear***function**,(*f*)=*x**mx***+**, is of the first degree;*b*its graph is a straight line with slope m,

the constant rate of change of f(x) (or y) with respect to x, and b is the line’s y-intercept.

A

*rational function*is of the form

where *P*(*x*) and *Q*(*x*) are polynomials.

The domain of

*f*is the set of all reals for which*Q*(*x*) ≠ 0.

The fundamental trigonometric identities, graphs, and reduction formulas are given in the Appendix.

The trigonometric functions are periodic.

A function

*f*is*periodic*__if there is a positive number__such that*p**f*(*x*+*p*) =*f*(*x*) for each*x*in the domain of*f*.The smallest such

*p*is called the*period***of**.*f*The graph of

*f*repeats every*p*units along the*x*-axis.The functions sin

*x*, cos*x*, csc*x*, and sec*x*have period 2π; tan*x*and cot*x*have period π.

Example:

SOLUTIONS:

We obtain

of the trigonometric functions by*inverses***limiting the domains of the latter,**so each trigonometric function is one-to-one over its restricted domain.

For example, we restrict

The inverse trigonometric function

**sin−1**is also commonly denoted by*x***arcsin**, which denotes*x**the*angle whose sine is*x*. The graph of sin−1*x*is, of course, the reflection of the graph of sin*x*in the line*y*=*x*.

The following laws of exponents hold for all rational *m* and *n*, provided that *a* > 0, *a* ≠ 1:

The

**exponential function***f*(*x*) =*ax*(*a*> 0,*a*≠ 1) is thus defined for all real*x*; its range is the__set of positive reals__.Of special interest and importance in calculus is the

**exponential function***f*(*x*) =*ex*, where*e*is an*irrational number*whose decimal approximation to five decimal places is 2.71828.

The domain of log

*ax*is the set of positive reals; its range is the set of all reals.It follows that the graphs of the pair of mutually inverse functions

*y*= 2*x*and*y*= log2*x*are symmetric to the line*y*=*x*

The logarithmic function log*a* *x* (*a* > 0, *a* ≠ 1) has the following properties:

The

**logarithmic base**is so important and convenient in calculus that we use a special symbol:*e*

log*e* *x* = ln *x*

Logarithms with base

*e*are called*natural***logarithms**. The**domain**__of ln__*x*__is the set of positive reals__; its__r____**ange**____is the set of all reals__. The graphs of the mutually inverse functions ln*x*and*ex*are given in the Appendix.

If the

*x*- and*y*-coordinates of a point on a graph are given as functions*f*and*g*of a third variable, say*t*, then

are called ** parametric equations** and

When

*t*represents time, as it often does, then we can view the curve as that followed by a moving particle as the time varies.

Example:

Find the Cartesian equation of, and sketch, the curve defined by the parametric equations

*x*=4 sin*t y*=5 cos *t* (0≦*t*≦2π)

SOLUTION:

**Polar coordinates** of the form (*r*,*θ*) identify the location of a point by specifying *θ*, an angle of rotation from the positive *x*-axis, and *r*, a distance from the origin.

A

**polar function**defines a curve with an equation of the form*r*=*f*(*θ*).

Some common polar functions include:

**Spiral**

**Rose**

**Cardioid**

**Limacon**

**Example**

Consider the polar function *r* = 2 + 4 sin *θ*.

(a)For what values of *θ* in the interval [0,2π] does the curve pass through the origin?

(b)For what value of *θ* in the interval [0,π/2] does the curve intersect the circle *r* = 3?

SOLUTION:

A

is a correspondence that associates with each element*function f*of a set*a*__called the__one and only one element*domain*of a set*b*__called the__*range**.*

We write ** f**(

**Inputs**- are the elements in the*domain*.**Outputs**- are the elements in the*range*.A

**function**is often represented by an equation, a graph, or a table.A

**vertical line**cuts the graph of a function in at most one point

**Example :**

Two functions

*f*and*g*with the same domain may be combined**to yield their sum and difference**:

*f*(*x*) + *g*(*x*) and *f*(*x*) − *g*(*x*), also written as (*f* + *g*) (*x*) and (*f* − *g*) (*x*)

or their product and quotient: *f*(*x*)*g*(*x*) and *f*(*x*)/*g*(*x*), also written as (*fg*) (*x*) and (*f* /*g*) (*x*)

The quotient is defined for all

*x*in the shared domain except those values for which*g*(*x*), the denominator, equals zero.

**Example:**

The

(or*composition**composite*) of*f*with*g*, written as*f*(*g*(*x*)) and read as “*f*of*g*of*x*,” is**the function obtained by replacing**wherever it occurs in*x**f*(*x*) by*g*(*x*). We also write (*f*∘*g*) (*x*) for*f*(*g*(*x*)).The domain of (

*f*∘*g*) (*x*) is the set of all*x*in the domain of*g*for which*g*(*x*) is in the domain of*f*.

**Example:**

**SOLUTION**:

The *absolute-value***function** *f*(*x*) = |*x*| and the *greatest-integer***function** *g*(*x*) = [*x*] are sketched in the Figure below:

**Example 8**

A function *f* is defined on the interval [−2,2] and has the graph shown in Figure 1-4.

(a)Sketch the graph of *y* = |*f*(*x*)|.

(b)Sketch the graph of *y* = *f*(|*x*|).

(c)Sketch the graph of *y* = −*f*(*x*).

(d)Sketch the graph of *y* = *f*(−*x*).

SOLUTIONS:

A

is of the form*polynomial function*

where ** n** is a

If *an* ≠ 0, the degree of the polynomial is *n*.

A

*linear***function**,(*f*)=*x**mx***+**, is of the first degree;*b*its graph is a straight line with slope m,

the constant rate of change of f(x) (or y) with respect to x, and b is the line’s y-intercept.

A

*rational function*is of the form

where *P*(*x*) and *Q*(*x*) are polynomials.

The domain of

*f*is the set of all reals for which*Q*(*x*) ≠ 0.

The fundamental trigonometric identities, graphs, and reduction formulas are given in the Appendix.

The trigonometric functions are periodic.

A function

*f*is*periodic*__if there is a positive number__such that*p**f*(*x*+*p*) =*f*(*x*) for each*x*in the domain of*f*.The smallest such

*p*is called the*period***of**.*f*The graph of

*f*repeats every*p*units along the*x*-axis.The functions sin

*x*, cos*x*, csc*x*, and sec*x*have period 2π; tan*x*and cot*x*have period π.

Example:

SOLUTIONS:

We obtain

of the trigonometric functions by*inverses***limiting the domains of the latter,**so each trigonometric function is one-to-one over its restricted domain.

For example, we restrict

The inverse trigonometric function

**sin−1**is also commonly denoted by*x***arcsin**, which denotes*x**the*angle whose sine is*x*. The graph of sin−1*x*is, of course, the reflection of the graph of sin*x*in the line*y*=*x*.

The following laws of exponents hold for all rational *m* and *n*, provided that *a* > 0, *a* ≠ 1:

The

**exponential function***f*(*x*) =*ax*(*a*> 0,*a*≠ 1) is thus defined for all real*x*; its range is the__set of positive reals__.Of special interest and importance in calculus is the

**exponential function***f*(*x*) =*ex*, where*e*is an*irrational number*whose decimal approximation to five decimal places is 2.71828.

The domain of log

*ax*is the set of positive reals; its range is the set of all reals.It follows that the graphs of the pair of mutually inverse functions

*y*= 2*x*and*y*= log2*x*are symmetric to the line*y*=*x*

The logarithmic function log*a* *x* (*a* > 0, *a* ≠ 1) has the following properties:

The

**logarithmic base**is so important and convenient in calculus that we use a special symbol:*e*

log*e* *x* = ln *x*

Logarithms with base

*e*are called*natural***logarithms**. The**domain**__of ln__*x*__is the set of positive reals__; its__r____**ange**____is the set of all reals__. The graphs of the mutually inverse functions ln*x*and*ex*are given in the Appendix.

If the

*x*- and*y*-coordinates of a point on a graph are given as functions*f*and*g*of a third variable, say*t*, then

are called ** parametric equations** and

When

*t*represents time, as it often does, then we can view the curve as that followed by a moving particle as the time varies.

Example:

Find the Cartesian equation of, and sketch, the curve defined by the parametric equations

*x*=4 sin*t y*=5 cos *t* (0≦*t*≦2π)

SOLUTION:

**Polar coordinates** of the form (*r*,*θ*) identify the location of a point by specifying *θ*, an angle of rotation from the positive *x*-axis, and *r*, a distance from the origin.

A

**polar function**defines a curve with an equation of the form*r*=*f*(*θ*).

Some common polar functions include:

**Spiral**

**Rose**

**Cardioid**

**Limacon**

**Example**

Consider the polar function *r* = 2 + 4 sin *θ*.

(a)For what values of *θ* in the interval [0,2π] does the curve pass through the origin?

(b)For what value of *θ* in the interval [0,π/2] does the curve intersect the circle *r* = 3?

SOLUTION: