Informative line

Other Types Of Functions

Practice trigonometric & trig functions formulas, six trigonometric functions, graphing trig functions & values and learn exponential and logarithmic functions, exponential function problems.

Logarithmic Function

  • \(f(x) = log_a \space x\)

         is called logarithmic function or simply log function, where 'a' is a positive constant.

  • If \(log_a \space x = y \) 
  • ⇒ \(x = a^y\)
  • Domain of \(f(x) =log_a \space x \space is \space (0, \infty)\)
  • Range of \(f(x) = log_a \space x \space is \space (-\infty, \infty)\)

Graph of \(log_a \space x = f(x) \)

Case 1 ( \(a>1\))

Note that shape is same for all \(a>1\) but an increase in the value of 'a' makes it increase more solwly.

Case 2 (\(0<a<1\))

Shape is same for all 'a' such that \(0<a<1\)

Illustration Questions

Which of the following is a logarithmic function?

A \(f(x) = 4^x\)

B \(f(x) = \Large \frac {sin \space x}{x + 1}\)

C \(f(x) = log_5 \space x\)

D \(f(x) = \Large \frac {x +1}{x^2 - 9}\)

×

\(f(x) = log_5 \space x\) is the only function of the form \(f(x) = log_a \space x\) (here base is 5)

Which of the following is a logarithmic function?

A

\(f(x) = 4^x\)

.

B

\(f(x) = \Large \frac {sin \space x}{x + 1}\)

C

\(f(x) = log_5 \space x\)

D

\(f(x) = \Large \frac {x +1}{x^2 - 9}\)

Option C is Correct

Root Function

  • In the power function

     \(f(x) = x^a \space if \space a = \large \frac {1}{n}\) where n is a positive integer then we say that it is a root function.

\(\therefore f(x) = x^{1/n} = \sqrt [n] x \space \text {is a root function}\)

  • n = 2, it is called the square root
  • n = 3, it is called the cube root
  • For even n, the domain of \(x^{1/n} \space is \space [0,\infty]\)
  • For odd n, the domain of \(x^{1/n} \space is \space (-\infty, \infty)\)

Graph of \(x^{1/n}\)

Case 1 :  n is even graph is similar to that of \(f(x)=\sqrt x\)

Case 2 :\( n\) is odd \((x>3)\) the graph is similar to \(f(x)=\sqrt [3] x\)

Illustration Questions

Which of the following will possibly represent graph of \(f(x) = x^{1/5}\)?

A

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B

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C

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D

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×

Graph of \(f(x) = x^{1/n}\) for odd n has a shape similar to that of \(f(x) = x^{1/3} \space or \space f(x) = \sqrt [3] {n}\)

image

Which of the following will possibly represent graph of \(f(x) = x^{1/5}\)?

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Option C is Correct

Reciprocal Function

  • In \(f(x) = x^a\) (power function) if we choose a = –1 we get \(f(x) = x^{-1} = \Large \frac {1}{x}\) which is called the reciprocal function.
  • The graph of reciprocal function is as shown.

\(\star\) Graph is entirely in the \(1^{st} \space \& \space 3^{rd}\) quadrant because x and y are of same sign if  \(y = 1/x.\)

\(\star\) Domain of reciprocal function is \(R - \{0\} \space or (-\infty, 0) \cup (0, \infty)\).

Illustration Questions

Which of the following is a point on the graph of a reciprocal function?

A (3, –5)

B (7, –8)

C (–2, –4)

D (1, –11)

×

In reciprocal function  

                                     \(y = 1/x\), so both x and y are of same sign for all points on the graph .So (–2, –4) is the required point

\(\star\) All point on \( y = 1/x\)  lie in \(1^{st} \space or \space 3^{rd} \space quadrant\)

Which of the following is a point on the graph of a reciprocal function?

A

(3, –5)

.

B

(7, –8)

C

(–2, –4)

D

(1, –11)

Option C is Correct

Rational Functions

  • Function of the form,  \(f(x) = \frac {P(x)}{Q(x)}\) where P(x) and Q(x) are polynomials in x, is called a rational function.
  • The domain of rational function is all values of x such that \(Q(x) \neq 0\)
  • Polynomials and reciprocal functions are special cases of Rational functions.

Illustration Questions

Which of the following is a rational function?

A \(\Large\frac {\sqrt {x} + 3}{x^2 + 7}\)

B \(\Large \frac {2x + 1}{x^2 - 4}\)

C \(\Large\frac {\sqrt [3]{x}}{3x + 4}\)

D \(x^{1/6}\)

×

Rational function is a ratio of two polynomial functions.

All options except 'b' have root function which are not allowed in rational functions.

Note that domain of \(f(x) = {\Large\frac {2x + 1}{x^2 - 4}} \space is \space \{x/x \neq \pm 2\}\)

Which of the following is a rational function?

A

\(\Large\frac {\sqrt {x} + 3}{x^2 + 7}\)

.

B

\(\Large \frac {2x + 1}{x^2 - 4}\)

C

\(\Large\frac {\sqrt [3]{x}}{3x + 4}\)

D

\(x^{1/6}\)

Option B is Correct

Algebraic Functions

  • Starting with polynomials, all functions which can be constructed using basic algebraic operations such as addition, subtraction, multiplication & division or taking roots, are called algebraic functions.
  • All Rational functions are algebraic functions.

e.g.

(1) \(f(x) = \sqrt {x^4 + x^2 + 1} \)

(2) \(f(x) = \dfrac {x^2 - 16x}{\sqrt x + 7}\)

Illustration Questions

Which of the following is not an algebraic function?

A \(f(x) = \Large \frac {x^2 - \sqrt 2}{2x + 3}\)

B \(f(x) = 3^x\)

C \(f(x) = x\sqrt {x + 7}\)

D \(f(x) = x^{1/4} \space (2 - x)\)

×

Note that Algebraic functions are made of polynomials and basic operations, option 'B' contains an exponential function which is not algebraic.

Which of the following is not an algebraic function?

A

\(f(x) = \Large \frac {x^2 - \sqrt 2}{2x + 3}\)

.

B

\(f(x) = 3^x\)

C

\(f(x) = x\sqrt {x + 7}\)

D

\(f(x) = x^{1/4} \space (2 - x)\)

Option B is Correct

Trigonometric Functions

There are six basic trigonometric functions

\((1) \space \text {f(x) = sin x}\)

\((2) \space \text {f(x) = cos x}\)

\((3) \space \text {f(x) = tan x} = \Large \frac {sin \space x}{cos \space x}\)

\((4) \space \text {f(x) = cot x} = \Large \frac {1}{tan \space x} = \frac {cos \space x}{sin \space x}\)

\((5) \space \text {f(x) = sec x} = \Large \frac {1}{cos \space x} \)

\((6) \space \text {f(x) = cosec x} = \Large \frac {1}{sin \space x} \)

Note that tan x, cot x, sec x and cosec x are defined in terms of sin x & cos x. So sin and cos are two basic trigonometric functions.

\(\star\) We assume that x is in radians unless otherwise mentioned

\(\star\) Domain of sin x & cos x is \(R \space or \space (-\infty, \infty)\)

\(\star\) Range of sin x & cos x is [–1, 1]

\(\star\) Roots of sin x = 0 is \(x = n \pi\) (n is an integer)

\(\star\) Roots of cos x = 0 is \(x = (2m + 1) \Large\frac {\pi}{2}\) (m is an integer)

\(\star\) All functions containing any of six basic Trigonometric functions and basic algebraic operations are in general called Trigonometric functions.

Graphs of f(x) = sin x

Graphs of f(x) = cos x

sin x & cos x are periodic with period \(2\pi\) and tan x is periodic with period \(\pi\)

Illustration Questions

Which of the following is a Trigonometric function?

A \(f(x) = \sqrt x\)

B \(f(x) = x^2\)

C \(f(x) = cos \space x\)

D \(f(x) = \Large \frac {\sqrt x}{x + 1}\)

×

\(f(x) = cos x\) is one of the six basic trigonometric functions.

Which of the following is a Trigonometric function?

A

\(f(x) = \sqrt x\)

.

B

\(f(x) = x^2\)

C

\(f(x) = cos \space x\)

D

\(f(x) = \Large \frac {\sqrt x}{x + 1}\)

Option C is Correct

Exponential Functions

  • The functions of the form \(f(x) = a^x\) where 'a' is a positive constant are called exponential functions.
  • The domain of exponential function is R.
  • The value of exponential function is \((0, \infty) \space or \space R^+\)

Graph:

Case 1 : ( \(a>1\))

If suppose \(f(x)=3^x\), if we increase the value of \(x\) the value of function increases. Also (0,1) is a part of all exponential function (all 'a')

Case 2 : ( \(0<a<1\))

If \(f(x)=(0.3)^x\) then increasing the value of \(x\) will decrease the value of the function. So graph is decreasing for \(0<a<1\)

Illustration Questions

Which of the following is an exponential function?

A \(f(x) = sin^2 \space x\)

B \(f(x) = \Large \frac {2x + 3}{x^2 - 9}\)

C \(f(x) = 7^x\)

D \(f(x) = \sqrt x\)

×

\(f(x) = 7^x\) is of the form \(f(x) = a^x\) where 'a' is 7.

Which of the following is an exponential function?

A

\(f(x) = sin^2 \space x\)

.

B

\(f(x) = \Large \frac {2x + 3}{x^2 - 9}\)

C

\(f(x) = 7^x\)

D

\(f(x) = \sqrt x\)

Option C is Correct

Kinds of Functions

There are many kinds of functions that are used to establish model or relationship in the world. We discuss the behaviour and graph of these functions, one by one

(1) Linear Function: The word LINEAR is derived from line, so graphs of linear functions are straight lines. These are functions of the form

                \(f(x) = mx + b\) (remember slope intercept form of a line)

               'm' is the slope and 'b' the y-intercept

               e.g.   \( y = 7x + 3\) → slope of this line is 7 and y-intercept is 3.

A function of the form

        \(f(x) = ax^2 + bx + c \) is called a quadratic function. Its graph is always of parabolic shape. The parabola opens upwards or downwards depending on whether a > 0 or a < 0.

A function P of the form
\(P(n) = a_n x^n + a_{n-1} x^{n-1}+ a_{n-2}x^{n-2} + .......... a_1x + a_0 \)

is called a polynomial function of degree n
(where n is non-negative integer) and \(a_n \neq 0\).

\(\star\) The domain of all polynomial function is \(R = (-\infty, \infty)\).

\(\star\) Linear and quadratic functions are special cases of polynomial functions with degree 1 and degree 2.

\(\star\) A polynomial of degree 3 is of the form

     \(f(x) = ax^3 + bx^2 + cx + d\) → It is called a cubic function.

\(\star\) Graph of some special cubics.

  • A function of the form

       \(f(x) = x^a\) is called power function.

      Here 'a' is any constant value. Depending on what value of 'a', we have different category of function.

  • If a = n where n is a positive integer.

        \(f(x) = x^n\) (polynomial with one term)

Note that graphs of \(x^2, x^4, x^6, ......, x^{24}\) one of similar shape whereas those of \(x^2, x^5, x^7, ......\) are similar. As n increases they become steeper.

  • In the power function

     \(f(x) = x^a \space if \space a = \large \frac {1}{n}\) where n is a positive integer then we say that it is a root function.

\(\therefore f(x) = x^{1/n} = \sqrt [n] x \space \text {is a root function}\)

  • n = 2, it is called the square root
  • n = 3, it is called the cube root

For even n, the domain of \(x^{1/n} \space is \space [0,\infty]\)

For odd n, the domain of \(x^{1/n} \space is \space (-\infty, \infty)\)

Graph of \(x^{1/n}\)

Case 1 :  n is even graph is similar to that of \(f(x)=\sqrt x\)

Case 2 :\( n\) is odd \((x>3)\) the graph is similar to \(f(x)=\sqrt [3] x\)

Function of the form, \(f(x) = \frac {P(x)}{Q(x)}\) where P(x) and Q(x) are polynomials in x, is called a rational function.

\(\star\) The domain of rational function is all values of x such that\(Q(x) \neq 0\)

\(\star\) Polynomials and reciprocal functions are special cases of Rational functions.

Starting with polynomials, all functions which can be constructed using basic algebraic operations such as addition, subtraction, multiplication & division or taking roots, are called algebraic functions.

\(\star\) All Rational functions are algebraic functions.

e.g.

(1) \(f(x) = \sqrt {x^4 + x^2 + 1} \)

(2) \(f(x) = \dfrac {x^2 - 16x}{\sqrt x + 7}\)

There are six basic trigonometric functions

\((1) \space \text {f(x) = sin x}\)

\((2) \space \text {f(x) = cos x}\)

\((3) \space \text {f(x) = tan x} = \Large \frac {sin \space x}{cos \space x}\)

\((4) \space \text {f(x) = cot x} = \Large \frac {1}{tan \space x} = \frac {cos \space x}{sin \space x}\)

\((5) \space \text {f(x) = sec x} = \Large \frac {1}{cos \space x} \)

\((6) \space \text {f(x) = cosec x} = \Large \frac {1}{sin \space x} \)

Note that tan x, cot x, sec x and cosec x are defined in terms of sin x & cos x. So sin and cos are two basic trigonometric functions.

\(\star\) We assume that x is in radians unless otherwise mentioned

\(\star\) Domain of sin x & cos x is \(R \space or \space (-\infty, \infty)\)

\(\star\) Range of sin x & cos x is [–1, 1]

\(\star\) Roots of sin x = 0 is \(x = n \pi\) (n is an integer)

\(\star\) Roots of cos x = 0 is \(x = (2m + 1) \Large\frac {\pi}{2}\) (m is an integer)

\(\star\) All functions contains any of six basic Trigonometric functions and basic algebraic operations are in general called Trigonometric functions.

Graphs of \(f(x) = sin x\)

Graphs of \(f(x) = cos x\)

\(sin \;x\) &\( cos\; x\) are periodic with period \(2\pi\) and tan x is periodic with period \(\pi\)

 

  • The functions of the form \(f(x) = a^x\) where 'a' is a positive constant are called exponential functions.
  • The domain of exponential function is R.
  • The value of exponential function is \((0, \infty) \space or \space R^+\)

Graph:

Case 1 : ( \(a>1\))

If suppose \(f(x)=3^x\), if we increase the value of \(x\) the value of function increases. Also (0,1) is a part of all exponential function (all 'a')

Case 2 : ( \(0<a<1\))

If \(f(x)=(0.3)^x\) then increasing the value of \(x\)will decrease the value of the function. So graph is decreasing for \(0<a<1\)

  • \(f(x) = log_a \space x\)

         is called logarithmic function or simply log function, where 'a' is a positive constant.

  • If \(log_a \space x = y \) 
  • ⇒ \(x = a^y\)
  • Domain of \(f(x) =log_a \space x \space is \space (0, \infty)\)
  • Range of \(f(x) = log_a \space x \space is \space (-\infty, \infty)\)

Graph of \(log_a \space x = f(x) \)

Case 1 ( \(a>1\))

Note that shape is same for all \(a>1\) but an increase in the value of 'a' makes it increase more slowly.

Case 2 (\(0<a<1\))

Shape is same for all 'a' such that \(0<a<1\)

Illustration Questions

The following options give a function with the name of the function against it. Choose the incorrect option.

A \(f(x)=log_5x\rightarrow\)logarithmic function

B \(f(x)=2^x\rightarrow\) exponential function

C \(f(x)=\dfrac {x-1}{x^2+x-3}\,\rightarrow\) rational function

D \(f(x)=sin^2x+3\,\rightarrow\) algebraic function

×

(1) Exponential functions are of the form \(\rightarrow a^x\)

(2) Logarithmic functions are of the form \(\rightarrow log_ax\)

(3) Rational functions are of the form \(\rightarrow \dfrac {P(x)}{Q(x)}\), where P and Q are polynomial.

(4) Algebraic function are formed by basic algebraic operation starting with polynomial.

Consider the option (A), (B), (C) are correct, the function in (D) is a trigonometric function.

\(\therefore \) (D) is incorrect.

The following options give a function with the name of the function against it. Choose the incorrect option.

A

\(f(x)=log_5x\rightarrow\)logarithmic function

.

B

\(f(x)=2^x\rightarrow\) exponential function

C

\(f(x)=\dfrac {x-1}{x^2+x-3}\,\rightarrow\) rational function

D

\(f(x)=sin^2x+3\,\rightarrow\) algebraic function

Option D is Correct

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